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A016038 -id:A016038 - OEIS
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Search: a016038 -id:a016038
Displaying 1-10 of 20 results found. page 1 2
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A050813 Numbers n not palindromic in any base b, 2 <= b <= 10. +10
10
19, 39, 47, 53, 58, 69, 75, 76, 79, 84, 87, 90, 94, 95, 96, 102, 103, 106, 108, 110, 115, 116, 120, 122, 132, 133, 134, 137, 139, 140, 143, 144, 147, 149, 152, 155, 158, 159, 163, 167, 168, 169, 174, 175, 176, 177, 179, 180, 183, 184, 187, 188, 193, 196, 198 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
T. D. Noe, Table of n, a(n) for n = 1..10000
P. De Geest, Palindromic numbers beyond base 10
FORMULA
A050812(n) = 0.
MATHEMATICA
n = -1; t = {}; While[Length[t] < 100, n++; If[Count[Table[s = IntegerDigits[n, m]; s == Reverse[s], {m, 2, 10}], True] == 0, AppendTo[t, n]]]; t (* T. D. Noe, Jul 18 2012 *)
CROSSREFS
Cf. A050812, A047811, A016038.
Cf. A214423, A214424, A214425, A214426 (palindromic in 1-4 bases).
KEYWORD
nonn,base
AUTHOR
Patrick De Geest, Oct 15 1999
STATUS
approved
A047811 Numbers n >= 4 that are not palindromic in any base b, 2 <= b <= n/2. +10
9
4, 6, 11, 19, 47, 53, 79, 103, 137, 139, 149, 163, 167, 179, 223, 263, 269, 283, 293, 311, 317, 347, 359, 367, 389, 439, 491, 563, 569, 593, 607, 659, 739, 827, 853, 877, 977, 983, 997, 1019, 1049, 1061, 1187, 1213, 1237, 1367, 1433, 1439, 1447, 1459 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Sequence A016038 is identical up to four additional terms: 0, 1, 2, 3; see there for more information.
Note that no prime p is palindromic in base b for the range sqrt(p) < b < p-1. Hence to find non-palindromic primes, we need only examine bases up to floor(sqrt(p)), which greatly reduces the computational effort required. - T. D. Noe, Mar 01 2008
This sequence is mentioned in the paper by Richard Guy, in which he reports on unsolved problems. This problem came from Mario Borelli and Cecil B. Mast. The paper poses two questions about these numbers: (1) Can palindromic or nonpalindromic primes be otherwise characterized? and (2) What is the cardinality, or the density, of the set of palindromic primes? Of the set of nonpalindromic primes? - T. D. Noe, Apr 17 2011
LINKS
T. D. Noe, Table of n, a(n) for n = 1..10000
R. K. Guy, Conway's RATS and other reversals, Amer. Math. Monthly, 96 (1989), 425-428.
FORMULA
a(n) = A016038(n+4) for all n. - M. F. Hasler, Sep 08 2015
MATHEMATICA
Select[Range[4, 1500], And@@(#!=Reverse[#]&/@Table[IntegerDigits[#, b], {b, 2, #/2}])&] (* Harvey P. Dale, May 22 2013 *)
PROG
(PARI) is(n)=!for(b=2, n\2, Vecrev(d=digits(n, b))==d&&return)&&n>3 \\ M. F. Hasler, Sep 08 2015
CROSSREFS
Cf. A016038, A050812, A050813.
Cf. A135549.
KEYWORD
nonn,base,easy,nice
AUTHOR
N. J. A. Sloane
EXTENSIONS
Extended (and corrected) by Patrick De Geest, Oct 15 1999
Minor edits by M. F. Hasler, Sep 08 2015
STATUS
approved
A065531 Number of palindromes in all base b representations for n, for 2<=b<=n. +10
9
1, 0, 1, 1, 2, 1, 2, 2, 2, 3, 1, 2, 2, 2, 3, 3, 3, 3, 1, 3, 4, 2, 2, 4, 2, 4, 3, 4, 2, 3, 3, 3, 3, 3, 2, 5, 2, 3, 2, 5, 2, 4, 2, 3, 4, 4, 1, 5, 2, 4, 4, 5, 1, 4, 4, 4, 4, 2, 2, 6, 2, 3, 5, 4, 5, 4, 3, 4, 2, 4, 2, 6, 3, 3, 3, 3, 2, 6, 1, 7, 3, 4, 2, 6, 5, 3, 2, 5, 2, 5, 4, 5, 4, 2, 2, 6, 2, 5, 4, 7, 2, 4, 1, 6, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
a(1) = 1 by convention, which makes this sequence different from A135551.
Index of first occurrence of k in A037183. - Robert G. Wilson v, Oct 27 2014
LINKS
Robert G. Wilson v, Table of n, a(n) for n = 1..1000
MATHEMATICA
palindromicBases[n_] := Module[{p}, Table[p = IntegerDigits[n, b]; If[ p == Reverse[p], {b, p}, Sequence @@ {}], {b, 2, n - 1}]]; Table[ Length[ palindromicBases[n]], {n, 105}] (* Robert G. Wilson v, May 12 2005 *)
CROSSREFS
Cf. A016038, A047811, A037183, A107129.
Essentially the same as A135551.
KEYWORD
easy,nonn,base
AUTHOR
Naohiro Nomoto, Dec 02 2001
STATUS
approved
A135549 Number of bases b, 1 < b < n-1, in which n is a palindrome. +10
9
0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 2, 0, 1, 1, 1, 2, 2, 2, 2, 0, 2, 3, 1, 1, 3, 1, 3, 2, 3, 1, 2, 2, 2, 2, 2, 1, 4, 1, 2, 1, 4, 1, 3, 1, 2, 3, 3, 0, 4, 1, 3, 3, 4, 0, 3, 3, 3, 3, 1, 1, 5, 1, 2, 4, 3, 4, 3, 2, 3, 1, 3, 1, 5, 2, 2, 2, 2, 1, 5, 0, 6, 2, 3, 1, 5, 4, 2, 1, 4, 1, 4, 3, 4, 3, 1, 1, 5, 1, 4, 3, 6, 1, 3, 0, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,11
COMMENTS
Every integer n is a palindrome when expressed in unary, or in base n-1 (where it will be 11). So here we assume 1 < b < n-1.
Records for a(n)>=1 are in A107129. - Dmitry Kamenetsky, Oct 22 2015
LINKS
T. D. Noe, Table of n, a(n) for n = 0..10000
John P. Linderman, Description of A135549-A135551 and A016038
John P. Linderman, Perl program [Use the command: palin.pl]
FORMULA
a(n) = A065531(n)-1 = A126071(n)-2 for n>2. - T. D. Noe, Feb 28 2008
MATHEMATICA
a = {0, 0, 0}; For[n = 4, n < 100, n++, c = 0; For[b = 2, b < n - 1, b++, If[IntegerDigits[n, b] == Reverse[IntegerDigits[n, b]], c++ ]]; AppendTo[a, c]]; a (* Stefan Steinerberger, Feb 27 2008 *)
Table[cnt=0; Do[d=IntegerDigits[n, b]; If[d==Reverse[d], cnt++ ], {b, 2, n-2}]; cnt, {n, 0, 100}] (* T. D. Noe, Feb 28 2008 *)
Table[Total[Boole[Table[PalindromeQ[IntegerDigits[n, b]], {b, 2, n-2}]]], {n, 0, 120}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 14 2020 *)
CROSSREFS
Cf. A016038, A037183, A065531, A107129, A135550, A135551.
Cf. A016038 (non-palindromic numbers in any base 1 < b < n-1)
KEYWORD
nonn,base
AUTHOR
John P. Linderman, Feb 26 2008, Feb 28 2008
STATUS
approved
A050812 Number of times n is palindromic in bases b, 2 <= b <= 10. +10
8
9, 9, 8, 8, 7, 7, 5, 5, 4, 3, 3, 1, 1, 1, 1, 2, 2, 2, 2, 0, 2, 3, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 0, 3, 1, 1, 1, 1, 2, 2, 0, 1, 1, 2, 2, 2, 0, 1, 3, 1, 2, 0, 1, 1, 1, 1, 3, 1, 3, 1, 2, 1, 0, 1, 1, 1, 2, 1, 0, 0, 1, 2, 0, 3, 1, 2, 1, 0, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
LINKS
T. D. Noe, Table of n, a(n) for n = 0..10000
Patrick De Geest, Palindromic numbers beyond base 10
EXAMPLE
a(121) = 4 since 121_10, 171_8, 232_7 and 11111_3 are palindromes.
MATHEMATICA
Table[Count[Table[IntegerDigits[n, b], {b, 2, 10}], _?(#==Reverse[#]&)], {n, 0, 90}] (* Harvey P. Dale, Aug 18 2012 *)
PROG
(Python)
from sympy.ntheory.digits import digits
def ispal(n, b):
digs = digits(n, b)[1:]
return digs == digs[::-1]
def a(n): return sum(ispal(n, b) for b in range(2, 11))
print([a(n) for n in range(87)]) # Michael S. Branicky, Sep 09 2021
(PARI) a(n) = sum(b=2, 10, my(d=digits(n, b)); d == Vecrev(d)); \\ Michel Marcus, Sep 09 2021
CROSSREFS
Cf. A050813, A016038, A002113.
KEYWORD
nonn,base
AUTHOR
Patrick De Geest, Oct 15 1999
STATUS
approved
A135551 Number of bases b, 1 < b < n, in which n is a palindrome. +10
8
0, 0, 0, 1, 1, 2, 1, 2, 2, 2, 3, 1, 2, 2, 2, 3, 3, 3, 3, 1, 3, 4, 2, 2, 4, 2, 4, 3, 4, 2, 3, 3, 3, 3, 3, 2, 5, 2, 3, 2, 5, 2, 4, 2, 3, 4, 4, 1, 5, 2, 4, 4, 5, 1, 4, 4, 4, 4, 2, 2, 6, 2, 3, 5, 4, 5, 4, 3, 4, 2, 4, 2, 6, 3, 3, 3, 3, 2, 6, 1, 7, 3, 4, 2, 6, 5, 3, 2, 5, 2, 5, 4, 5, 4, 2, 2, 6, 2, 5, 4, 7, 2, 4, 1, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,6
COMMENTS
Every integer n is a palindrome when expressed in unary, or in base n-1 (where it will be 11).
First occurrence in A037183.
a(n) is always less than A001221(n) except for 2 and 6; a(n) is always less than A001222(n) except for even powers of twos and 6, 12, 81, 243, 625, 729, 2187, 19683, 59049, ..., . - Robert G. Wilson v, Jul 17 2016
LINKS
Robert G. Wilson v, Table of n, a(n) for n = 0..100000
John P. Linderman, Description of A135549-A135551 and A016038
John P. Linderman, Perl program [Use the command: BASEDELTA=0 palin.pl]
FORMULA
a(n) = A135549(n) + 1 for n>2; otherwise a(n) = A135549(n) = 0. - Michel Marcus, Oct 15 2014
a(n) = A126071(n) - 1. - Michel Marcus, Mar 07 2015
MATHEMATICA
palindromicBases[n_] := Module[{p}, Table[p = IntegerDigits[n, b]; If[p == Reverse[p], {b, p}, Sequence @@ {}], {b, 2, n - 1}]]; Array[ Length@ palindromicBases@# &, 105, 0] (* Robert G. Wilson v, Oct 15 2014 *)
palQ[n_Integer, base_Integer] := Module[{idn = IntegerDigits[n, base]}, idn == Reverse[idn]];
f[n_] := Block[{s = Ceiling@ Sqrt@ n, b = 2, c = If[ IntegerQ@ Sqrt[4n + 1], -1, 0]}, While[b < s, If[ palQ[n, b], c++]; b++]; c + Count[ Mod[n, Range[s - 1]], 0]]; f[0] = 0; Array[f, 105, 0] (* much faster for large Ns *) (* Robert G. Wilson v, Oct 20 2014 *)
CROSSREFS
Cf. A037183, A135549, A135550, A016038.
Essentially the same as A065531.
KEYWORD
nonn,base
AUTHOR
John P. Linderman, Feb 26 2008, Feb 28 2008
STATUS
approved
A016026 Smallest base relative to which n is palindromic. +10
7
2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 10, 5, 3, 6, 2, 3, 2, 5, 18, 3, 2, 10, 3, 5, 4, 3, 2, 3, 4, 9, 2, 7, 2, 4, 6, 5, 6, 4, 12, 3, 5, 4, 6, 10, 2, 4, 46, 7, 6, 7, 2, 3, 52, 8, 4, 3, 5, 28, 4, 9, 6, 5, 2, 7, 2, 10, 5, 3, 22, 9, 7, 5, 2, 6, 14, 18, 10, 5, 78, 3, 8, 3, 5, 11, 2, 6, 28, 5, 8, 14, 3, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
From Hieronymus Fischer, Jan 05 2014: (Start)
The terms are well defined since each number m > 2 is palindromic in base m - 1.
A number n > 6 is prime, if a(n) = n - 1.
Numbers m of the form m = q * p with q < p - 1, are palindromic in base p - 1, and therefore a(m) <= p.
Numbers m of the form m := j*(p^k - 1)/(p - 1), 1 <= j < p are palindromic in base p, and therefore: a(m) <= p.(End)
LINKS
Hieronymus Fischer, Table of n, a(n) for n = 1..10000 (first 1000 from Vincenzo Librandi).
K. S. Brown, On General Palindromic Numbers
FORMULA
From Hieronymus Fischer, Jan 05 2014: (Start)
a(A016038(n)) = A016038(n) - 1, for n > 3.
a(A006995(n)) = 2, for n > 1.
a(A002113(n)) <= 10 for n > 1. (End)
To put Fischer's comments in words: if n > 3 is a strictly non-palindromic number (A016038), then a(n) = n - 1. If n > 1 is a binary palindrome (A006995), then a(n) = 2. And if n > 1 is a decimal palindrome, then a(n) <= 10. - Alonso del Arte, Sep 15 2017
EXAMPLE
n = 4 = 11_3 is palindromic in base 3, but not palindromic in base 2, hence a(4) = 3. [Typo corrected by Phil Ronan, May 22 2014]
n = 14 = 22_6 is palindromic in base 6, but not palindromic in any other base < 6, hence a(14) = 6.
MATHEMATICA
palQ[n_, b_] := Reverse[x = IntegerDigits[n, b]] == x; Table[base = 2; While[!palQ[n, base], base++]; base, {n, 92}] (* Jayanta Basu, Jul 26 2013 *)
PROG
(PARI) ispal(n, b) = my(d=digits(n, b)); d == Vecrev(d);
a(n) = my(b=2); while (! ispal(n, b), b++); b; \\ Michel Marcus, Sep 22 2017
(Python)
from itertools import count
from sympy.ntheory.factor_ import digits
def A016026(n): return next(b for b in count(2) if (s := digits(n, b)[1:])[:(t:=len(s)+1>>1)]==s[:-t-1:-1]) # Chai Wah Wu, Jan 17 2024
CROSSREFS
Cf. A016038, A006995, A002113.
KEYWORD
nonn,base
AUTHOR
Robert G. Wilson v
STATUS
approved
A139819 Complement of repdigit numbers A010785. +10
7
10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 89 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Identical to (base 10) non-palindromic numbers A029742 up to a(83) = 101 which is a term of this sequence but not in A029742. - M. F. Hasler, Sep 08 2015
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
E. W. Weisstein, Repdigit, World of Mathematics, wolfram.com.
Wikipedia, Repdigit
FORMULA
A202022(a(n)) = 0. - Reinhard Zumkeller, Dec 09 2011
MAPLE
isA139819 := proc(n)
convert(n, base, 10) ;
convert(%, set) ;
simplify(nops(%) >1 ) ;
end proc: # R. J. Mathar, Jan 17 2017
PROG
(Haskell) a139819 n = a139819_list !! (n-1)
a139819_list = filter ((== 0) . a202022) [0..] -- Reinhard Zumkeller, Dec 09 2011
(PARI) is_A139819(n)=#Set(digits(n))>1 \\ M. F. Hasler, Sep 08 2015
CROSSREFS
Cf. A066484 (subsequence).
Cf. A029742 (non-palindromic in base 10), A016038 (in any base), A050813 (in bases 2..10).
KEYWORD
nonn,base
AUTHOR
N. J. A. Sloane, Jun 02 2008
STATUS
approved
A123586 Numbers that are not palindromes of 3 or more digits in some base b >= 2. +10
4
1, 2, 3, 4, 6, 8, 11, 12, 14, 18, 19, 22, 24, 30, 32, 35, 39, 44, 47, 48, 53, 54, 58, 60, 66, 69, 70, 75, 76, 77, 79, 84, 87, 90, 94, 95, 96, 102, 103, 106, 108, 110, 115, 116, 120, 132, 134, 137, 139, 140, 143, 147, 149, 152, 158, 159, 163, 167, 168, 174, 175, 176 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Table of n, a(n) for n=1..62.
CROSSREFS
Cf. A114255 (complement), A016038.
KEYWORD
base,easy,nonn
AUTHOR
Franklin T. Adams-Watters, Nov 13 2006
STATUS
approved
A126071 Number of bases (2 <= b <= n+1) in which n is a palindrome. +10
4
1, 1, 2, 2, 3, 2, 3, 3, 3, 4, 2, 3, 3, 3, 4, 4, 4, 4, 2, 4, 5, 3, 3, 5, 3, 5, 4, 5, 3, 4, 4, 4, 4, 4, 3, 6, 3, 4, 3, 6, 3, 5, 3, 4, 5, 5, 2, 6, 3, 5, 5, 6, 2, 5, 5, 5, 5, 3, 3, 7, 3, 4, 6, 5, 6, 5, 4, 5, 3, 5, 3, 7, 4, 4, 4, 4, 3, 7, 2, 8, 4, 5, 3, 7, 6, 4, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
a(n) >= 1, since n will always have a single "digit" in base n+1.
LINKS
T. D. Noe, Table of n, a(n) for n = 1..1000
EXAMPLE
From bases 2 to 9 respectively, 8 can be represented as: 1000, 22, 20, 13, 12, 11, 10, 8. Three of those are symmetrical (22, 11, 8) and so a(8) = 3.
MATHEMATICA
Table[cnt = 0; Do[d = IntegerDigits[n, k]; If[d == Reverse[d], cnt++], {k, 2, n + 1}]; cnt, {n, 100}] (* T. D. Noe, Oct 04 2012 *)
PROG
(PARI) a(n) = sum(k=2, n+1, d = digits(n, k); Vecrev(d) == d); \\ Michel Marcus, Mar 07 2015
CROSSREFS
Cf. A016026.
Cf. A016038, A047811 (related to numbers having 2 bases).
KEYWORD
nonn,base
AUTHOR
Paul Richards, Mar 01 2007
EXTENSIONS
Extended by T. D. Noe, Oct 04 2012
STATUS
approved
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