(Translated by https://www.hiragana.jp/)
A055023 -id:A055023 - OEIS
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
Search: a055023 -id:a055023
Displaying 1-8 of 8 results found. page 1
     Sort: relevance | references | number | modified | created      Format: long | short | data
A323070 Numbers k such that A055023(k) != A323071(k), where A323071(k) = gcd(k, 1+A060681(k)). +20
2
55, 105, 155, 203, 253, 355, 405, 455, 497, 595, 655, 689, 705, 737, 755, 791, 955, 979, 1005, 1027, 1055, 1081, 1221, 1255, 1305, 1355, 1379, 1555, 1605, 1655, 1673, 1703, 1711, 1751, 1855, 1905, 1955, 1967, 2065, 2155, 2189, 2205, 2255, 2261, 2329, 2455, 2505, 2555, 2755, 2805, 2849, 2855, 3055 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Equivalently, numbers k for which A055032(k) != A323072(k).
Neither primes nor prime powers present?
LINKS
Table of n, a(n) for n=1..53.
PROG
(PARI)
A055023(n) = (n/denominator((sum(m=1, n - 1, m^(n - 1)) + 1)/n)); \\ From A055023.
A060681(n) = (n-if(1==n, n, n/vecmin(factor(n)[, 1])));
A323071(n) = gcd(n, 1+A060681(n));
is_A323070(n) = (A055023(n)!=A323071(n));
CROSSREFS
Cf. A055023, A055032, A060681, A323071, A323072.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jan 04 2019
STATUS
approved
A055030 (Sum(m^(p-1),m=1..p-1)+1)/p as p runs through the primes. +10
11
1, 2, 71, 9596, 1355849266, 1032458258547, 1653031004194447737, 3167496749732497119310, 22841077183004879532481321652, 1768861419039838982256898243427529138091, 10293527624511391856267274608237685758691696 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
It is conjectured that (Sum(m^(n-1),m=1..n-1)+1)/n is an integer iff n is 1 or a prime.
Always an integer from little Fermat theorem. Converse is conjectured to be true: if p | (1+1^(p-1)+2^(p-1)+3^(p-1)+...+(p-1)^(p-1)) and p > 1, then p is prime. That was checked by Giuga up to p <= 10^1000. [Benoit Cloitre, Jun 09 2002]
For Sum(m^p, m=1..p-1)/p as p runs through the odd primes, see A219550. - Jonathan Sondow, Oct 31 2017
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, A17.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..76
K. MacMillan and J. Sondow, Proofs of power sum and binomial coefficient congruences via Pascal's identity, Amer. Math. Monthly, 118 (2011), 549-551.
FORMULA
a(n) = (1+A225578(n))/A000040(n). - R. J. Mathar, Jan 09 2017
MAPLE
A055030 := proc(n)
p := ithprime(n) ;
add(m^(p-1), m=1..p-1) ;
(1+%)/p ;
end proc:
seq(A055030(n), n=1..5) ; # R. J. Mathar, Jan 09 2017
MATHEMATICA
Array[(Sum[m^(# - 1), {m, # - 1}] + 1)/# &@ Prime@ # &, 11] (* Michael De Vlieger, Nov 04 2017 *)
PROG
(PARI) for(n=1, 20, print1((1+sum(i=1, prime(n)-1, i^(prime(n)-1)))/prime(n), ", ")) /* Benoit Cloitre, Jun 09 2002*/
CROSSREFS
Cf. A055031, A055032, A055023, A201560, A204187, A219550, A294507.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jun 11 2000
EXTENSIONS
Comments corrected by Jonathan Sondow, Jan 11 2012
STATUS
approved
A204187 a(n) = Sum_{m=1..n-1} m^(n-1) modulo n. +10
10
0, 1, 2, 0, 4, 3, 6, 0, 6, 5, 10, 0, 12, 7, 10, 0, 16, 9, 18, 0, 14, 11, 22, 0, 20, 13, 18, 0, 28, 15, 30, 0, 22, 17, 0, 0, 36, 19, 26, 0, 40, 21, 42, 0, 21, 23, 46, 0, 42, 25, 34, 0, 52, 27, 0, 0, 38, 29, 58, 0, 60, 31, 42, 0, 52, 33, 66, 0, 46, 35, 70, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
a(n) = n - 1 if n is 1 or a prime, by Fermat's little theorem. It is conjectured that the converse is also true; see A055032 and A201560 and note that a(n) = n-1 <==> A055032(n) = 1 <==> A201560(n) = 0.
As of 1991, Giuga and Bedocchi had verified no composite n < 10^1700 satisfies a(n) = n - 1 (Ribemboim, 1991). - Alonso del Arte, May 10 2013
REFERENCES
Steve Dinh, The Hard Mathematical Olympiad Problems And Their Solutions, AuthorHouse, 2011, Problem 6 of Hong Kong Mathematical Olympiad 2007 (find a(7)), page 134.
Richard K. Guy, Unsolved Problems in Number Theory, A17.
Paulo Ribemboim, The Little Book of Big Primes. New York: Springer-Verlag (1991): 17.
LINKS
Ivan Neretin, Table of n, a(n) for n = 1..10000
John Clark, On a conjecture involving Fermat's Little Theorem, Thesis, 2008, University of South Florida.
Hong Kong Mathematics Olympiad (2007-2008), Final Event 2 (Group), problem 2, p. 437.
Bernard Schott, Proof that a(n)=n/2 if n is of the form 4k+2.
Index to sequences related to Olympiads.
FORMULA
a(p) = p - 1 if p is prime, and a(4n) = 0.
a(n) + 1 == A201560(n) (mod n).
a(n) = n/2 iff n is of the form 4k+2 (conjectured). - Ivan Neretin, Sep 23 2016
a(4*k+2) = 2*k+1; for a proof see corresponding link. - Bernard Schott, Dec 29 2021
EXAMPLE
Sum(m^3, m = 1 .. 3) = 1^3 + 2^3 + 3^3 = 36 == 0 (mod 4), so a(4) = 0.
MATHEMATICA
Table[Mod[Sum[i^(n - 1), {i, n - 1}], n], {n, 75}] (* Alonso del Arte, May 10 2013 *)
PROG
(PARI) a(n) = lift(sum(i=1, n, Mod(i, n)^(n-1))); \\ Michel Marcus, Feb 23 2020
(Python)
def a(n): return sum(pow(m, n-1, n) for m in range(1, n))%n
print([a(n) for n in range(1, 73)]) # Michael S. Branicky, Jan 02 2022
CROSSREFS
Cf. A055023, A055030, A055031, A055032, A201560.
Cf. A191677 (zeros).
KEYWORD
nonn,easy
AUTHOR
Jonathan Sondow, Jan 12 2012
STATUS
approved
A055032 Denominator of (Sum(m^(n-1),m=1..n-1)+1)/n. +10
9
1, 1, 1, 4, 1, 3, 1, 8, 9, 5, 1, 12, 1, 7, 15, 16, 1, 9, 1, 20, 7, 11, 1, 24, 25, 13, 27, 28, 1, 15, 1, 32, 33, 17, 35, 36, 1, 19, 13, 40, 1, 21, 1, 44, 45, 23, 1, 48, 49, 25, 51, 52, 1, 27, 55, 56, 19, 29, 1, 60, 1, 31, 63, 64, 65, 33, 1, 68, 69, 35, 1, 72, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
It is conjectured that this is 1 iff n is 1 or a prime.
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, A17.
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
MAPLE
a:= proc(n) local S, m;
S:= 1;
for m from 1 to n-1 do
S:= S + m &^(n-1) mod n;
od:
denom(S/n);
end proc;
seq(a(n), n=1..1000); # Robert Israel, May 30 2014
MATHEMATICA
Table[Denominator[(Sum[m^(n - 1), {m, 1, n - 1}] + 1)/n], {n, 1, 10}] (* G. C. Greubel, Jun 06 2016 *)
PROG
(PARI) a(n) = denominator((sum(m=1, n - 1, m^(n - 1)) + 1)/n); \\ Indranil Ghosh, May 17 2017
(Python)
from sympy import Integer
def a(n): return ((sum(m**(n - 1) for m in range(1, n)) + 1)/Integer(n)).denominator() # Indranil Ghosh, May 17 2017
CROSSREFS
Cf. A055031, A055030, A055023, A201560, A204187.
KEYWORD
nonn,frac
AUTHOR
N. J. A. Sloane, Jun 11 2000
STATUS
approved
A055031 Numerator of (Sum(m^(n-1),m=1..n-1)+1)/n. +10
6
1, 1, 2, 37, 71, 2213, 9596, 1200305, 24684613, 287152493, 1355849266, 427675990237, 1032458258547, 228796942438201, 16841089312342856, 665478473553144001, 1653031004194447737, 631449646252135295657, 3167496749732497119310 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
REFERENCES
R. K. Guy, Unsolved Problems Number Theory, A17.
LINKS
Indranil Ghosh, Table of n, a(n) for n = 1..200
MATHEMATICA
Table[Numerator[(Sum[m^(n - 1), {m, n - 1}] + 1)/n], {n, 50}] (* Indranil Ghosh, May 17 2017 *)
PROG
(PARI) a(n) = numerator((sum(m=1, n - 1, m^(n - 1)) + 1)/n); \\ Indranil Ghosh, May 17 2017
(Python)
from sympy import Integer
def a(n): return ((sum(m**(n - 1) for m in range(1, n)) + 1)/Integer(n)).numerator() # Indranil Ghosh, May 17 2017
CROSSREFS
Cf. A055032, A055030, A055023, A201560, A204187.
KEYWORD
nonn,frac
AUTHOR
N. J. A. Sloane, Jun 11 2000
STATUS
approved
A201560 a(n) = (Sum(m^(n-1), m=1..n-1) + 1) modulo n. +10
6
0, 0, 0, 1, 0, 4, 0, 1, 7, 6, 0, 1, 0, 8, 11, 1, 0, 10, 0, 1, 15, 12, 0, 1, 21, 14, 19, 1, 0, 16, 0, 1, 23, 18, 1, 1, 0, 20, 27, 1, 0, 22, 0, 1, 22, 24, 0, 1, 43, 26, 35, 1, 0, 28, 1, 1, 39, 30, 0, 1, 0, 32, 43, 1, 53, 34, 0, 1, 47, 36, 0, 1, 0, 38, 51, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,6
COMMENTS
Equals 0 if n is 1 or a prime, by Fermat's little theorem. It is conjectured that the converse is also true; see A055030, A055032, A204187 and note that a(n) = 0 <==> A055032(n) = 1 <==> A204187(n) = n-1.
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, A17.
LINKS
Ivan Neretin, Table of n, a(n) for n = 1..10000
FORMULA
a(prime) = 0 and a(4n) = 1.
a(n) == A204187(n) + 1 (mod n).
EXAMPLE
Sum(m^3, m=1..3) + 1 = 1^3 + 2^3 + 3^3 + 1 = 37 == 1 (mod 4), so a(4) = 1.
MATHEMATICA
Table[Mod[Plus @@ PowerMod[Range[n - 1], n - 1, n] + 1, n], {n, 77}] (* Ivan Neretin, Sep 23 2016 *)
CROSSREFS
Cf. A055023, A055030, A055031, A055032, A204187.
KEYWORD
nonn
AUTHOR
Jonathan Sondow, Jan 11 2012
STATUS
approved
A323071 a(n) = gcd(n, 1+A060681(n)). +10
5
1, 2, 3, 1, 5, 2, 7, 1, 1, 2, 11, 1, 13, 2, 1, 1, 17, 2, 19, 1, 3, 2, 23, 1, 1, 2, 1, 1, 29, 2, 31, 1, 1, 2, 1, 1, 37, 2, 3, 1, 41, 2, 43, 1, 1, 2, 47, 1, 1, 2, 1, 1, 53, 2, 5, 1, 3, 2, 59, 1, 61, 2, 1, 1, 1, 2, 67, 1, 1, 2, 71, 1, 73, 2, 3, 1, 1, 2, 79, 1, 1, 2, 83, 1, 1, 2, 1, 1, 89, 2, 1, 1, 3, 2, 1, 1, 97, 2, 1, 1, 101, 2, 103, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Differs from A055023 at n = 55, 105, 155, ..., (A323070).
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
FORMULA
a(n) = gcd(n, 1+A060681(n)).
a(n) = n/A323072(n).
PROG
(PARI)
A060681(n) = (n-if(1==n, n, n/vecmin(factor(n)[, 1])));
A323071(n) = gcd(n, 1+A060681(n));
CROSSREFS
Cf. A055023, A060681, A323070, A323072.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jan 04 2019
STATUS
approved
A340078 a(n) = gcd(n, 1+A018804(n)), where A018804(n) = Sum_{k=1..n} gcd(k, n). +10
3
1, 2, 3, 1, 5, 2, 7, 1, 1, 2, 11, 1, 13, 2, 1, 1, 17, 2, 19, 1, 3, 2, 23, 1, 1, 2, 1, 7, 29, 2, 31, 1, 1, 2, 1, 1, 37, 2, 3, 1, 41, 14, 43, 1, 5, 2, 47, 1, 1, 2, 1, 1, 53, 2, 5, 1, 3, 2, 59, 1, 61, 2, 1, 1, 1, 2, 67, 1, 1, 2, 71, 1, 73, 2, 1, 1, 1, 2, 79, 1, 1, 2, 83, 1, 1, 2, 1, 1, 89, 2, 1, 1, 3, 2, 1, 3, 97, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..8191
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
FORMULA
a(n) = gcd(n, 1+A018804(n)).
PROG
(PARI)
A018804(n) = sumdiv(n, d, n*eulerphi(d)/d); \\ From A018804
A340078(n) = gcd(n, 1+A018804(n));
CROSSREFS
Cf. A018804, A340079, A340080.
Cf. also A055023, A323071 (similar but different sequences).
KEYWORD
nonn
AUTHOR
Antti Karttunen, Dec 30 2020
STATUS
approved
page 1

Search completed in 0.006 seconds

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 7 13:22 EDT 2024. Contains 375730 sequences. (Running on oeis4.)