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A004817 -id:A004817 - OEIS
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Search: a004817 -id:a004817
Displaying 1-10 of 30 results found. page 1 2 3
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A003072 Numbers that are the sum of 3 positive cubes. +10
85
3, 10, 17, 24, 29, 36, 43, 55, 62, 66, 73, 80, 81, 92, 99, 118, 127, 129, 134, 136, 141, 153, 155, 160, 179, 190, 192, 197, 216, 218, 225, 232, 244, 251, 253, 258, 270, 277, 281, 288, 307, 314, 342, 344, 345, 349, 352, 359, 368, 371, 375, 378, 397, 405, 408, 415, 433, 434 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
A119977 is a subsequence; if m is a term then there exists at least one k>0 such that m-k^3 is a term of A003325. - Reinhard Zumkeller, Jun 03 2006
A025456(a(n)) > 0. - Reinhard Zumkeller, Apr 23 2009
Davenport proved that a(n) << n^(54/47 + e) for every e > 0. - Charles R Greathouse IV, Mar 26 2012
LINKS
K. D. Bajpai, Table of n, a(n) for n = 1..12955 (first 1000 terms from T. D. Noe)
H. Davenport, Sums of three positive cubes, J. London Math. Soc., 25 (1950), 339-343. Coll. Works III p. 999.
Eric Weisstein's World of Mathematics, Cubic Number
Index entries for sequences related to sums of cubes
FORMULA
{n: A025456(n) >0}. - R. J. Mathar, Jun 15 2018
EXAMPLE
a(11) = 73 = 1^3 + 2^3 + 4^3, which is sum of three cubes.
a(15) = 99 = 2^3 + 3^3 + 4^3, which is sum of three cubes.
MAPLE
isA003072 := proc(n)
local x, y, z;
for x from 1 do
if 3*x^3 > n then
return false;
end if;
for y from x do
if x^3+2*y^3 > n then
break;
end if;
if isA000578(n-x^3-y^3) then
return true;
end if;
end do:
end do:
end proc:
for n from 1 to 1000 do
if isA003072(n) then
printf("%d, ", n) ;
end if;
end do: # R. J. Mathar, Jan 23 2016
MATHEMATICA
Select[Range[435], (p = PowersRepresentations[#, 3, 3]; (Select[p, #[[1]] > 0 && #[[2]] > 0 && #[[3]] > 0 &] != {})) &] (* Jean-François Alcover, Apr 29 2011 *)
With[{upto=500}, Select[Union[Total/@Tuples[Range[Floor[Surd[upto-2, 3]]]^3, 3]], #<=upto&]] (* Harvey P. Dale, Oct 25 2021 *)
PROG
(PARI) sum(n=1, 11, x^(n^3), O(x^1400))^3 /* Then [i|i<-[1..#%], polcoef(%, i)] gives the list of powers with nonzero coefficient. - M. F. Hasler, Aug 02 2020 */
(PARI) list(lim)=my(v=List(), k, t); lim\=1; for(x=1, sqrtnint(lim-2, 3), for(y=1, min(sqrtnint(lim-x^3-1, 3), x), k=x^3+y^3; for(z=1, min(sqrtnint(lim-k, 3), y), listput(v, k+z^3)))); Set(v) \\ Charles R Greathouse IV, Sep 14 2015
(Haskell)
a003072 n = a003072_list !! (n-1)
a003072_list = filter c3 [1..] where
c3 x = any (== 1) $ map (a010057 . fromInteger) $
takeWhile (> 0) $ map (x -) $ a003325_list
-- Reinhard Zumkeller, Mar 24 2012
CROSSREFS
Subsequence of A004825.
Cf. A003325, A024981, A057904 (complement), A010057, A000578, A023042 (subsequence of cubes).
Cf. A###### (x, y) = Numbers that are the sum of x nonzero y-th powers: A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane, David W. Wilson
EXTENSIONS
Incorrect program removed by David A. Corneth, Aug 01 2020
STATUS
approved
A003336 Numbers that are the sum of 2 positive 4th powers. +10
77
2, 17, 32, 82, 97, 162, 257, 272, 337, 512, 626, 641, 706, 881, 1250, 1297, 1312, 1377, 1552, 1921, 2402, 2417, 2482, 2592, 2657, 3026, 3697, 4097, 4112, 4177, 4352, 4721, 4802, 5392, 6497, 6562, 6577, 6642, 6817, 7186, 7857, 8192, 8962, 10001, 10016, 10081, 10256, 10625 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Numbers k such that k = x^4 + y^4 has a solution in positive integers x, y.
There are no squares in this sequence. - Altug Alkan, Apr 08 2016
As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020
LINKS
Sean A. Irvine, Table of n, a(n) for n = 1..20000 (terms 1..1000 from T. D. Noe, terms 1001..10000 from David A. Corneth)
A. Bremner and P. Morton, A new characterization of the integer 5906, Manuscripta Math. 44 (1983) 187-229; Math. Rev. 84i:10016.
S. R. Finch, On a generalized Fermat-Wiles equation [broken link]
Steven R. Finch, On  Generalized Fermat-Wiles Equation [From the Wayback Machine]
Samuel S. Wagstaff, Jr., Equal Sums of Two Distinct Like Powers, J. Int. Seq., Vol. 25 (2022), Article 22.3.1.
Eric Weisstein's World of Mathematics, Biquadratic Number.
FORMULA
{i: A216284(i) > 0}. - R. J. Mathar, Jun 04 2021
EXAMPLE
From David A. Corneth, Aug 01 2020: (Start)
16378801 is in the sequence as 16378801 = 43^4 + 60^4.
39126977 is in the sequence as 39126977 = 49^4 + 76^4.
71769617 is in the sequence as 71769617 = 19^4 + 92^4. (End)
MATHEMATICA
nn=12; Select[Union[Plus@@@(Tuples[Range[nn], {2}]^4)], # <= nn^4&] (* Harvey P. Dale, Dec 29 2010 *)
Select[Range@ 11000, Length[PowersRepresentations[#, 2, 4] /. {0, _} -> Nothing] > 0 &] (* Michael De Vlieger, Apr 08 2016 *)
PROG
(PARI) list(lim)=my(v=List()); for(x=1, sqrtnint(lim\=1, 4), for(y=1, min(sqrtnint(lim-x^4, 4), x), listput(v, x^4+y^4))); Set(v) \\ Charles R Greathouse IV, Apr 24 2012; updated July 13 2024
(PARI) T=thueinit('x^4+1, 1);
is(n)=#thue(T, n)>0 && !issquare(n) \\ Charles R Greathouse IV, Feb 26 2017
(Python)
def aupto(lim):
p1 = set(i**4 for i in range(1, int(lim**.25)+2) if i**4 <= lim)
p2 = set(a+b for a in p1 for b in p1 if a+b <= lim)
return sorted(p2)
print(aupto(10625)) # Michael S. Branicky, Mar 18 2021
CROSSREFS
5906 is the first term in A060387 but not in this sequence. Cf. A020897.
Cf. A088687 (2 distinct 4th powers).
A###### (x, y): Numbers that are the form of x nonzero y-th powers.
Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).
Cf. A000583 (4th powers).
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A003327 Numbers that are the sum of 4 positive cubes in 1 or more way. +10
57
4, 11, 18, 25, 30, 32, 37, 44, 51, 56, 63, 67, 70, 74, 81, 82, 88, 89, 93, 100, 107, 108, 119, 126, 128, 130, 135, 137, 142, 144, 145, 149, 154, 156, 161, 163, 168, 180, 182, 187, 191, 193, 198, 200, 205, 206, 217, 219, 224, 226, 233, 240, 243, 245, 252, 254 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
It is conjectured that every number greater than 7373170279850 is in this sequence. [See the paper of the same name. - T. D. Noe, May 25 2017] - Charles R Greathouse IV, Jan 14 2017
As the order of addition doesn't matter we can assume terms are in increasing order. - David A. Corneth, Aug 01 2020
LINKS
David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Jean-Marc Deshouillers, François Hennecart, Bernard Landreau, 7373170279850, Math. Comp. 69 (2000), pp. 421-439. Appendix by I. Gusti Putu Purnaba.
Eric Weisstein's World of Mathematics, Cubic Number.
EXAMPLE
From David A. Corneth, Aug 01 2020: (Start)
3888 is in the sequence as 3888 = 6^3 + 6^3 + 12^3 + 12^3.
7729 is in the sequence as 7729 = 2^3 + 4^3 + 14^3 + 17^3.
7875 is in the sequence as 7875 = 5^3 + 10^3 + 15^3 + 15^3. (End)
PROG
(PARI) list(lim)=my(v=List(), e=1+lim\1, x='x, t); t=sum(i=1, sqrtnint(e-4, 3), x^i^3, O(x^e))^4; for(n=4, lim, if(polcoeff(t, n)>0, listput(v, n))); Vec(v) \\ Charles R Greathouse IV, Jan 14 2017
CROSSREFS
Cf. A025403, A057905 (complement), A025411 (distinct).
A###### (x, y): Numbers that are the form of x nonzero y-th powers.
Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
EXTENSIONS
More terms from Eric W. Weisstein
STATUS
approved
A000414 Numbers that are the sum of 4 nonzero squares. +10
51
4, 7, 10, 12, 13, 15, 16, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
As the order of addition doesn't matter we can assume terms are in increasing order. - David A. Corneth, Aug 01 2020
LINKS
David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Index entries for sequences related to sums of squares
FORMULA
a(n) = n + O(log n). - Charles R Greathouse IV, Sep 03 2014
EXAMPLE
From David A. Corneth, Aug 01 2020: (Start)
1608 is in the sequence as 1608 = 18^2 + 20^2 + 20^2 + 22^2.
2140 is in the sequence as 2140 = 21^2 + 21^2 + 23^2 + 27^2.
3298 is in the sequence as 3298 = 25^2 + 26^2 + 29^2 + 34^2. (End)
MATHEMATICA
q=16; lst={}; Do[Do[Do[Do[z=a^2+b^2+c^2+d^2; If[z<=(q^2)+3, AppendTo[lst, z]], {d, q}], {c, q}], {b, q}], {a, q}]; Union@lst (*Vladimir Joseph Stephan Orlovsky, Feb 07 2010 *)
PROG
(PARI) is(n)=my(k=if(n, n/4^valuation(n, 4), 2)); k!=2 && k!=6 && k!=14 && !setsearch([0, 1, 3, 5, 9, 11, 17, 29, 41], n) \\ Charles R Greathouse IV, Sep 03 2014
(Python)
limit = 10026 # 10000th term in b-file
from functools import lru_cache
nzs = [k*k for k in range(1, int(limit**.5)+2) if k*k + 3 <= limit]
nzss = set(nzs)
@lru_cache(maxsize=None)
def ok(n, m): return n in nzss if m == 1 else any(ok(n-s, m-1) for s in nzs)
print([n for n in range(4, limit+1) if ok(n, 4)]) # Michael S. Branicky, Apr 07 2021
(Python)
from itertools import count, islice
def A000414_gen(startvalue=0): # generator of terms >= startvalue
return filter(lambda n:not(n in {0, 1, 3, 5, 9, 11, 17, 29, 41} or n>>((~n&n-1).bit_length()&-2) in {2, 6, 14}), count(max(startvalue, 0)))
A000414_list = list(islice(A000414_gen(), 30)) # Chai Wah Wu, Jul 09 2022
CROSSREFS
Cf. A000534 (complement).
A###### (x, y): Numbers that are the form of x nonzero y-th powers.
Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane and J. H. Conway
EXTENSIONS
corrected 6/95
STATUS
approved
A003328 Numbers that are the sum of 5 positive cubes. +10
49
5, 12, 19, 26, 31, 33, 38, 40, 45, 52, 57, 59, 64, 68, 71, 75, 78, 82, 83, 89, 90, 94, 96, 97, 101, 108, 109, 115, 116, 120, 127, 129, 131, 134, 135, 136, 138, 143, 145, 146, 150, 152, 153, 155, 157, 162, 164, 169, 171, 172, 176, 181, 183, 188, 190, 192, 194, 195, 199 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
As the order of addition doesn't matter we can assume terms are in increasing order. - David A. Corneth, Aug 01 2020
It seems only a finite number N of positive integers are not in this sequence, and thus a(n) = n - N for all sufficiently large n. Is it true that 2243453, last term of A048927, is sufficiently large in that sense? - M. F. Hasler, Jan 04 2023
LINKS
David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Cubic Number.
EXAMPLE
From David A. Corneth, Aug 01 2020: (Start)
3084 is in the sequence as 3084 = 5^3 + 5^3 + 5^3 + 8^3 + 13^3.
4385 is in the sequence as 4385 = 4^3 + 4^3 + 9^3 + 11^3 + 13^3.
5426 is in the sequence as 5426 = 8^3 + 9^3 + 9^3 + 12^3 + 12^3. (End)
PROG
(PARI) select( {is_A003328(n, k=5, m=3, L=sqrtnint(abs(n-k+1), m))=if( n>k*L^m || n<k, 0, n<k*L^m, forstep(r=min(k-1, n\L^m), 0, -1, self()(n-r*L^m, k-r, m, L-1) && return(1)), 1)}, [1..200]) \\ M. F. Hasler, Aug 02 2020
A003328_upto(N, k=5, m=3)=[i|i<-[1..#N=sum(n=1, sqrtnint(N, m), 'x^n^m, O('x^N))^k], polcoef(N, i)] \\ M. F. Hasler, Aug 02 2020
(Python)
from collections import Counter
from itertools import combinations_with_replacement as combs_w_rep
def aupto(lim):
s = filter(lambda x: x<=lim, (i**3 for i in range(1, int(lim**(1/3))+2)))
s2 = filter(lambda x: x<=lim, (sum(c) for c in combs_w_rep(s, 5)))
s2counts = Counter(s2)
return sorted(k for k in s2counts)
print(aupto(200)) # Michael S. Branicky, May 12 2021
CROSSREFS
Cf. A003328, A048926, A048297.
Cf. A057906 (Complement)
Cf. A###### (x, y) = Numbers that are the sum of x nonzero y-th powers:
A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A003337 Numbers n which are the sum of 3 nonzero 4th powers. +10
46
3, 18, 33, 48, 83, 98, 113, 163, 178, 243, 258, 273, 288, 338, 353, 418, 513, 528, 593, 627, 642, 657, 707, 722, 768, 787, 882, 897, 962, 1137, 1251, 1266, 1298, 1313, 1328, 1331, 1378, 1393, 1458, 1506, 1553, 1568, 1633, 1808, 1875, 1922, 1937, 2002, 2177 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Numbers which are in this sequence but not in A047714 must also be the sum of 2 biquadrates, or equal to a fourth power. Among the first 1000 terms of this sequence, this is the case for 4802 = 2*7^4, 57122 = 2*13^4 and 76832 = 2*14^4. - M. F. Hasler, Dec 31 2012
The union of A047714, A336536, and fourth powers of A003294. - Robert Israel, Jul 24 2020
As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020
LINKS
David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Biquadratic Number.
EXAMPLE
From David A. Corneth, Aug 01 2020: (Start)
194818 is in the sequence as 194818 = 3^4 + 4^4 + 21^4.
480113 is in the sequence as 480113 = 7^4 + 12^4 + 26^4.
693842 is in the sequence as 693842 = 13^4 + 15^4 + 28^4. (End)
PROG
(Python)
def aupto(lim):
p1 = set(i**4 for i in range(1, int(lim**.25)+2) if i**4 <= lim)
p2 = set(a+b for a in p1 for b in p1 if a+b <= lim)
p3 = set(apb+c for apb in p2 for c in p1 if apb+c <= lim)
return sorted(p3)
print(aupto(2400)) # Michael S. Branicky, Mar 18 2021
CROSSREFS
Cf. A003294, A047714, A336536, A309762, A344188.
A###### (x, y): Numbers that are the form of x nonzero y-th powers.
Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A003358 Numbers that are the sum of 2 nonzero 6th powers. +10
44
2, 65, 128, 730, 793, 1458, 4097, 4160, 4825, 8192, 15626, 15689, 16354, 19721, 31250, 46657, 46720, 47385, 50752, 62281, 93312, 117650, 117713, 118378, 121745, 133274, 164305, 235298, 262145, 262208, 262873, 266240, 277769, 308800, 379793, 524288 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020
LINKS
David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Samuel S. Wagstaff, Jr., Equal Sums of Two Distinct Like Powers, J. Int. Seq., Vol. 25 (2022), Article 22.3.1.
EXAMPLE
From David A. Corneth, Aug 01 2020: (Start)
10069120217 is in the sequence as 10069120217 = 29^6 + 46^6.
139314070233 is in the sequence as 139314070233 = 3^6 + 72^6.
404680615040 is in the sequence as 404680615040 = 22^6 + 86^6. (End)
MATHEMATICA
With[{k = 6}, Union@ Map[(#[[1]]^k + #[[2]]^k) &, Tuples[Range[8], {2}]]] (* Michael De Vlieger, Sep 09 2022, after Harvey P. Dale at A004999 *)
CROSSREFS
Cf. A088677 (2 distinct 6th). Supersequence of A106318.
A###### (x, y): Numbers that are the form of x nonzero y-th powers.
Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
EXTENSIONS
Removed incorrect program. David A. Corneth, Aug 01 2020
STATUS
approved
A003380 Numbers that are the sum of 2 nonzero 8th powers. +10
44
2, 257, 512, 6562, 6817, 13122, 65537, 65792, 72097, 131072, 390626, 390881, 397186, 456161, 781250, 1679617, 1679872, 1686177, 1745152, 2070241, 3359232, 5764802, 5765057, 5771362, 5830337, 6155426, 7444417, 11529602, 16777217, 16777472, 16783777, 16842752 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
David A. Corneth, Table of n, a(n) for n = 1..10000 (first 5833 terms from R. J. Mathar, replacing an earlier b-file that was missing terms)
EXAMPLE
From David A. Corneth, Aug 01 2020: (Start)
274893519322337 is in the sequence as 274893519322337 = 58^8 + 59^8.
357707312890625 is in the sequence as 357707312890625 = 50^8 + 65^8.
2590188068194497 is in the sequence as 2590188068194497 = 57^8 + 84^8. (End)
MAPLE
A003380 := proc(nmax::integer)
local a, x, x8, y, y8 ;
a := {} ;
for x from 1 do
x8 := x^8 ;
if 2*x8 > nmax then
break;
end if;
for y from x do
y8 := y^8 ;
if x8+y8 > nmax then
break;
end if;
if x8+y8 <= nmax then
a := a union {x8+y8} ;
end if;
end do:
end do:
sort(convert(a, list)) ;
end proc:
nmax := 20000000000000000 ;
L:= A003380(nmax) ;
LISTTOBFILE(L, "b003380.txt", 1) ; # R. J. Mathar, Aug 01 2020
MATHEMATICA
Total/@Tuples[Range[8]^8, 2]//Union (* Harvey P. Dale, Apr 04 2017 *)
PROG
(PARI) list(lim)=my(v=List(), x8); for(x=1, sqrtnint(lim\=1, 8), x8=x^8; for(y=1, min(sqrtnint(lim-x8, 8), x), listput(v, x8+y^8))); Set(v) \\ Charles R Greathouse IV, Aug 22 2017
CROSSREFS
Subsequence of A004875.
Cf. A155468 (2 distinct 8th).
A###### (x, y): Numbers that are the form of x nonzero y-th powers.
Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A003335 Numbers that are the sum of 12 positive cubes. +10
40
12, 19, 26, 33, 38, 40, 45, 47, 52, 54, 59, 61, 64, 66, 68, 71, 73, 75, 78, 80, 82, 85, 87, 89, 90, 92, 94, 96, 97, 99, 101, 103, 104, 106, 108, 110, 111, 113, 115, 116, 117, 118, 120, 122, 123, 124, 125, 127, 129, 130, 131, 132, 134, 136, 137, 138, 139, 141, 142 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020
LINKS
David A. Corneth, Table of n, a(n) for n = 1..10000
EXAMPLE
From David A. Corneth, Aug 01 2020: (Start)
1120 is in the sequence as 1120 = 2^3 + 3^3 + 4^3 + 4^3 + 4^3 + 4^3 + 4^3 + 4^3 + 4^3 + 5^3 + 8^3.
2339 is in the sequence as 2339 = 4^3 + 4^3 + 4^3 + 4^3 + 5^3 + 5^3 + 5^3 + 5^3 + 5^3 + 9^3 + 9^3.
3594 is in the sequence as 3594 = 4^3 + 5^3 + 6^3 + 6^3 + 6^3 + 6^3 + 7^3 + 7^3 + 7^3 + 8^3 + 10^3. (End)
PROG
(PARI) (A003335_upto(N, k=12, m=3)=[i|i<-[1..#N=sum(n=1, sqrtnint(N, m), 'x^n^m, O('x^N))^k], polcoef(N, i)])(150) \\ Use 2nd & 3rd optional arg to get other sequences of this family. See A003333 for alternate code. - M. F. Hasler, Aug 03 2020
CROSSREFS
A###### (x, y): Numbers that are the form of x nonzero y-th powers.
Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).
Cf. A000578 (cubes).
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A003338 Numbers that are the sum of 4 nonzero 4th powers. +10
40
4, 19, 34, 49, 64, 84, 99, 114, 129, 164, 179, 194, 244, 259, 274, 289, 304, 324, 339, 354, 369, 419, 434, 499, 514, 529, 544, 594, 609, 628, 643, 658, 673, 674, 708, 723, 738, 769, 784, 788, 803, 849, 868, 883, 898, 913, 963, 978, 1024, 1043, 1138, 1153, 1218 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020
LINKS
David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Biquadratic Number.
EXAMPLE
From David A. Corneth, Aug 01 2020: (Start)
53667 is in the sequence as 53667 = 2^4 + 5^4 + 7^4 + 15^4.
81427 is in the sequence as 81427 = 5^4 + 5^4 + 11^4 + 16^4.
106307 is in the sequence as 106307 = 3^4 + 5^4 + 5^4 + 18^4. (End)
MAPLE
# returns number of ways of writing n as a^4+b^4+c^4+d^4, 1<=a<=b<=c<=d.
A003338 := proc(n)
local a, i, j, k, l, res ;
a := 0 ;
for i from 1 do
if i^4 > n then
break ;
end if;
for j from i do
if i^4+j^4 > n then
break ;
end if;
for k from j do
if i^4+j^4+k^4> n then
break;
end if;
res := n-i^4-j^4-k^4 ;
if issqr(res) then
res := sqrt(res) ;
if issqr(res) then
l := sqrt(res) ;
if l >= k then
a := a+1 ;
end if;
end if;
end if;
end do:
end do:
end do:
a ;
end proc:
for n from 1 do
if A003338(n) > 0 then
print(n) ;
end if;
end do: # R. J. Mathar, May 17 2023
MATHEMATICA
f[maxno_]:=Module[{nn=Floor[Power[maxno-3, 1/4]], seq}, seq=Union[Total/@(Tuples[Range[nn], {4}]^4)]; Select[seq, #<=maxno&]]
f[1000] (* Harvey P. Dale, Feb 27 2011 *)
PROG
(Python)
limit = 1218
from functools import lru_cache
qd = [k**4 for k in range(1, int(limit**.25)+2) if k**4 + 3 <= limit]
qds = set(qd)
@lru_cache(maxsize=None)
def findsums(n, m):
if m == 1: return {(n, )} if n in qds else set()
return set(tuple(sorted(t+(q, ))) for q in qds for t in findsums(n-q, m-1))
print([n for n in range(4, limit+1) if len(findsums(n, 4)) >= 1]) # Michael S. Branicky, Apr 19 2021
CROSSREFS
Cf. A047715, A309763 (more than 1 way), A344189 (exactly 2 ways), A176197 (distinct nonzero powers).
A###### (x, y): Numbers that are the form of x nonzero y-th powers.
Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
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