A016038 -id:A016038 - cp's OEIS frontend

A065531 Number of palindromes in all base b representations for n, for 2<=b<=n.

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

Offset: 1

Naohiro Nomoto, Dec 02 2001

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

Robert G. Wilson v, Table of n, a(n) for n = 1..1000

Cf. A016038, A047811, A037183, A107129.

Essentially the same as A135551.

f = Compile[{{n, Integer}}, Module[{b = 2, c = Floor@ If[ IntegerQ@ Sqrt[n + 1], -1, 0], idn, s = Floor@ Sqrt[n + 1] - 1}, c = c + Count[ Mod[n, Range@ s], 0]; s = s + 2; While[b < s, If[ Reverse[idn = IntegerDigits[n, b]] == idn, c++]; b++]; c]]; f[1] = 1; Array[f, 105] (* _Robert G. Wilson v, May 05 2025 *)

A135549 Number of bases b, 1 < b < n-1, in which n is a palindrome.

Original entry on oeis.org

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

Offset: 0

John P. Linderman, Feb 26 2008, Feb 28 2008

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

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]

Cf. A016038, A037183, A065531, A107129, A135550, A135551.

Cf. A016038 (non-palindromic numbers in any base 1 < b < n-1)

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 *)

a(n) = A065531(n)-1 = A126071(n)-2 for n>2. - T. D. Noe, Feb 28 2008

A050813 Numbers n not palindromic in any base b, 2 <= b <= 10.

Original entry on oeis.org

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

Offset: 1

Patrick De Geest, Oct 15 1999

T. D. Noe, Table of n, a(n) for n = 1..10000
P. De Geest, Palindromic numbers beyond base 10

Cf. A050812, A047811, A016038.

Cf. A214423, A214424, A214425, A214426 (palindromic in 1-4 bases).

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 *)

A050812(n) = 0.

A135551 Number of bases b, 1 < b < n, in which n is a palindrome.

Original entry on oeis.org

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

Offset: 0

John P. Linderman, Feb 26 2008, Feb 28 2008

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

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]

Cf. A037183, A135549, A135550, A016038.

Essentially the same as A065531.

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 *)

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

A047811 Numbers n >= 4 that are not palindromic in any base b, 2 <= b <= n/2.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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

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.

Cf. A016038, A050812, A050813.

Cf. A135549.

Select[Range[4,1500],And@@(#!=Reverse[#]&/@Table[IntegerDigits[#,b],{b,2,#/2}])&] (* Harvey P. Dale, May 22 2013 *)

is(n)=!for(b=2,n\2,Vecrev(d=digits(n,b))==d&&return)&&n>3 \\ M. F. Hasler, Sep 08 2015

a(n) = A016038(n+4) for all n. - M. F. Hasler, Sep 08 2015

Extended (and corrected) by Patrick De Geest, Oct 15 1999

Minor edits by M. F. Hasler, Sep 08 2015

A050812 Number of times n is palindromic in bases b, 2 <= b <= 10.

Original entry on oeis.org

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

Offset: 0

Patrick De Geest, Oct 15 1999

			a(121) = 4 since 121_10, 171_8, 232_7 and 11111_3 are palindromes.

T. D. Noe, Table of n, a(n) for n = 0..10000
Patrick De Geest, Palindromic numbers beyond base 10

Cf. A050813, A016038, A002113.

Table[Count[Table[IntegerDigits[n,b],{b,2,10}],?(#==Reverse[#]&)],{n,0,90}] (* _Harvey P. Dale, Aug 18 2012 *)

a(n) = sum(b=2, 10, my(d=digits(n, b)); d == Vecrev(d)); \\ Michel Marcus, Sep 09 2021

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

A016026 Smallest base relative to which n is palindromic.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v

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)

			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.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000 (first 1000 from Vincenzo Librandi).
K. S. Brown, On General Palindromic Numbers

Cf. A016038, A006995, A002113.

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 *)

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

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

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

A139819 Complement of repdigit numbers A010785.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jun 02 2008

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Repdigit.
Wikipedia, Repdigit
Chai Wah Wu, Algorithms for complementary sequences, arXiv:2409.05844 [math.NT], 2024.

Cf. A066484 (subsequence).

Cf. A029742 (non-palindromic in base 10), A016038 (in any base), A050813 (in bases 2..10).

a139819 n = a139819_list !! (n-1)
a139819_list = filter ((== 0) . a202022) [0..] -- Reinhard Zumkeller, Dec 09 2011

isA139819 := proc(n)
    convert(n,base,10) ;
    convert(%,set) ;
    simplify(nops(%) >1 ) ;
end proc: # R. J. Mathar, Jan 17 2017

is_A139819(n)=#Set(digits(n))>1 \\ M. F. Hasler, Sep 08 2015

def A139819(n):
    m, k = n, n+9*((l:=len(str(n)))-1)+9*n//(10**l-1)
    while m != k:
        m, k = k, n+9*((l:=len(str(k)))-1)+9*k//(10**l-1)
    return m # Chai Wah Wu, Sep 04 2024

A202022(a(n)) = 0. - Reinhard Zumkeller, Dec 09 2011

A126071 Number of bases (2 <= b <= n+1) in which n is a palindrome.

Original entry on oeis.org

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

Offset: 1

Paul Richards, Mar 01 2007

a(n) >= 1, since n will always have a single "digit" in base n+1.

			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.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A016026.

Cf. A016038, A047811 (related to numbers having 2 bases).

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 *)
a = Compile[{{n, Integer}}, Module[{b = 2, c = Floor@ If[ IntegerQ@ Sqrt[n + 1], 0, 1], idn, s = Floor@ Sqrt[n + 1] - 1}, c = c + Count[ Mod[n, Range@ s], 0]; s = s + 2; While[ b < s, If[ Reverse[ idn = IntegerDigits[n, b]] == idn, c++]; b++]; c]]; Array[a, 87] (* _Robert G. Wilson v, May 05 2025 *)

a(n) = sum(k=2, n+1, d = digits(n, k); Vecrev(d) == d); \\ Michel Marcus, Mar 07 2015

Extended by T. D. Noe, Oct 04 2012

A123586 Numbers that are not palindromes of 3 or more digits in some base b >= 2.

Original entry on oeis.org

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

Offset: 1

Franklin T. Adams-Watters, Nov 13 2006

Cf. A114255 (complement), A016038.

cp's OEIS Frontend

A065531 Number of palindromes in all base b representations for n, for 2<=b<=n.

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A135549 Number of bases b, 1 < b < n-1, in which n is a palindrome.

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A050813 Numbers n not palindromic in any base b, 2 <= b <= 10.

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A135551 Number of bases b, 1 < b < n, in which n is a palindrome.

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A047811 Numbers n >= 4 that are not palindromic in any base b, 2 <= b <= n/2.

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A050812 Number of times n is palindromic in bases b, 2 <= b <= 10.

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A016026 Smallest base relative to which n is palindromic.

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A139819 Complement of repdigit numbers A010785.

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