A016026 -id:A016026 - cp's OEIS frontend

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

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

A249634 Least number k that is a palindrome in base n but no bases less than n, or 0 if no such k exists.

Original entry on oeis.org

0, 1, 2, 25, 6, 14, 32, 54, 30, 11, 84, 39, 140, 75, 176, 102, 198, 19, 220, 147, 110, 69, 384, 175, 416, 486, 420, 58, 570, 279, 544, 429, 306, 245, 684, 296, 380, 663, 880, 615, 1134, 258, 1012, 1035, 1104, 47, 1392, 539, 1500, 1071, 1508, 53, 2106, 935, 1736, 1311, 1798, 413, 2940, 671

Offset: 1

Robert G. Wilson v, Nov 02 2014

This sequence gives the first occurrence of n in A016026.

"Of course, every positive integer has a palindromic representation in SOME base. If we let f(n) denote the smallest base relative to which n is palindromic, then clearly f(n) is no greater than n-1, because every number n has the palindromic form '11' in the base (n-1)." [See Math Pages link; f(n)=A016026(n).]

			a(6) = 14 because 14_10 equals 22_6. And 14 is the least integer whose representation in base 6 yields a palindrome as its first palindrome. 7, though palindromic in base 6, is also palindromic in a base less than 6 (7_10 = 111_2 = 11_6) so 7 cannot be a(6).

Robert G. Wilson v, Table of n, a(n) for n = 1..1000
Math Pages, On General Palindromic Numbers

Cf. A016026.

N:= 100: # to get a(1) to a(N)
ispali:= proc(k,b) local L; L:= convert(k,base,b); L = ListTools:-Reverse(L); end proc:
Needed:= N-1:
for k from 1 while Needed > 0 do
   for b from 2 to N while not ispali(k,b) do od:
   if b <= N and not assigned(A[b]) then A[b]:= k; Needed:= Needed - 1 fi
od:
0, seq(A[n],n=1..N); # Robert Israel, Nov 04 2014

f[n_] := Block[{b = 2}, While[ Reverse[idn = IntegerDigits[n, b]] != idn, b++]; b]; a = Array[f, 3000]; Table[ Position[a, n, 1, 1], {n, 2, 60}] // Flatten

a(n)=m=1;while(m,c=0;for(k=2,n-1,D=digits(m,k);if(D==Vecrev(D),c++;break));if(!c&&(d=digits(m,n))==Vecrev(d),return(m));m++)
print1(0,", ");for(n=2,100,print1(a(n),", ")) \\ Derek Orr, Nov 02 2014

A086757 Smallest prime p such that n is a palindrome in base-p representation.

Original entry on oeis.org

2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 13, 5, 3, 13, 2, 3, 2, 5, 23, 3, 2, 23, 3, 5, 29, 3, 2, 3, 31, 29, 2, 7, 2, 37, 37, 5, 41, 37, 41, 3, 5, 13, 47, 43, 2, 5, 53, 7, 53, 7, 2, 3, 59, 17, 59, 3, 5, 59, 61, 11, 67, 5, 2, 7, 2, 67, 5, 3, 71, 13, 7, 5, 2, 73, 79, 37, 79, 5, 83, 3, 83, 3, 5, 11, 2

Offset: 1

Reinhard Zumkeller, Aug 01 2003

A016026(n) <= a(n) <= A007918(n).

Cf. A007918, A016026.

Cf. A006995 (a(n)=2).

isok(p,n) = my(d=digits(n,p)); d == Vecrev(d);
a(n) = my(p=2); while (!isok(p,n), p=nextprime(p+1)); p; \\ Michel Marcus, Jan 30 2024

from sympy import sieve
from sympy.ntheory import is_palindromic
def a086757(n): return next(p for p in sieve if is_palindromic(n, p)) # Dumitru Damian, Jan 29 2024

A065437 Smallest base relative to which the n-th prime is palindromic.

Original entry on oeis.org

3, 2, 2, 2, 10, 3, 2, 18, 3, 4, 2, 6, 5, 6, 46, 52, 4, 6, 5, 7, 2, 78, 5, 8, 8, 10, 102, 2, 5, 8, 2, 10, 136, 138, 148, 3, 7, 162, 166, 3, 178, 10, 6, 12, 6, 11, 8, 222, 8, 12, 3, 14, 12, 8, 2, 262, 268, 7, 11, 14, 282, 292, 7, 310, 2, 316, 15, 9, 346, 8, 10, 358, 366, 4, 13, 10, 388

Offset: 1

Peter Bertok (peter(AT)bertok.com), Nov 23 2001

Subset of A016026 for primes only.

			71 is the 20th prime and can be written as 131 in base 7, hence a(20)=7.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Palindromic Primes

Cf. A016026.

PrimeMinBase[ n_ ] := NestWhile[ # + 1 &, 2, IntegerDigits[ Prime[ n ], # ] != Reverse[ IntegerDigits[ Prime[ n ], # ] ] & ]
sbr[n_]:=Module[{k=2},While[IntegerDigits[n,k]!=Reverse[ IntegerDigits[ n,k]], k++]; k]; Table[sbr[n],{n,Prime[Range[80]]}] (* Harvey P. Dale, Jun 07 2016 *)

A065809 a(n) is the smallest number m > n such that m is palindromic in base n and is not palindromic in bases b with 2 <= b < n.

Original entry on oeis.org

3, 4, 25, 6, 14, 32, 54, 30, 11, 84, 39, 140, 75, 176, 102, 198, 19, 220, 147, 110, 69, 384, 175, 416, 486, 420, 58, 570, 279, 544, 429, 306, 245, 684, 296, 380, 663, 880, 615, 1134, 258, 1012, 1035, 1104, 47, 1392, 539, 1500, 1071, 1508, 53, 2106

Offset: 2

Naohiro Nomoto, Dec 06 2001

Index at which first occurrence of n occurs in A016026 when the palindrome is multidigit. Only the first two terms of A016026 are single-digit palindromes. - Robert G. Wilson v, Dec 22 2021

			From _Robert G. Wilson v_, Dec 22 2021: (Start)
a(2) = 3 since A016026(3) = 2;
a(3) = 4 since A016026(4) = 3;
a(4) = 25 since A016026(25) = 4; etc. (End)

Robert G. Wilson v, Table of n, a(n) for n = 2..10000 (first 441 terms from Robert Israel).

Cf. A016026, A065531, A249634.

N:= 100:
f:= proc(m) local k,L;
   for k from 2 to m do
     L:= convert(m,base,k);
     if L = ListTools:-Reverse(L) then return k fi
   od;
   FAIL
end proc:
V:= Array(2..N):
count:= 0:
for m from 3 while count < N-1 do
  v:= f(m);
  if v = FAIL or v > N then next fi;
  if V[v] = 0 then count:= count+1; V[v]:= m;
  fi
od:
convert(V,list); # Robert Israel, Apr 08 2019

fpb = Compile[{{n, Integer}}, Module[{b = 2, idn}, While[ Reverse[idn = IntegerDigits[n, b]] != idn, b++]; b]]; k = 3; t[] := 0; While[k < 2200, a = fpb@ k; If[ t[a] == 0, t[a] = k]; k++]; t@# & /@ Range[2, 53] (* Robert G. Wilson v, Dec 22 2021 *)

Definition edited by N. J. A. Sloane, Apr 08 2019

A369233 Smallest base for which the digits expansion of 2^n is palindromic.

Original entry on oeis.org

3, 3, 3, 3, 7, 7, 7, 15, 7, 7, 31, 7, 15, 15, 31, 15, 15, 31, 63, 15, 31, 31, 127, 63, 31, 31, 63, 127, 127, 31, 63, 63, 255, 255, 127, 63, 63, 127, 511, 255, 255, 63, 127, 127, 511, 511, 511, 255, 127, 127, 255, 1023, 1023, 511, 511, 127, 255, 255, 2047, 1023, 1023, 1023, 127, 255

Offset: 1

Michel Marcus, Jan 17 2024

From the Kreher and Stinson article we know that a(n) is of the form 2^k-1 (cf. A000225). - David A. Corneth, Jan 18 2024

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 400 terms from Michel Marcus)
Donald L. Kreher and Douglas R. Stinson, On min-base palindromic representations of powers of 2, arXiv:2401.07351 [math.NT], 2024. See Table 3 p. 7.

Cf. A000079, A000225, A016026.

f:= proc(n) local x,b,L,i;
  x:= 2^n;
  for b from 3 do
    L:= convert(x,base,b);
    if andmap(i -> L[i]=L[-i], [$1..nops(L)/2]) then return b fi
  od
end proc:
map(f,[$1..100]); # Robert Israel, Jan 17 2024

A369233[n_] := Block[{p = 2^n, k = 1}, While[!PalindromeQ[IntegerDigits[p, 2^++k-1]]]; 2^k-1]; Array[A369233, 100] (* Paolo Xausa, Mar 10 2024 *)

ispal(n, b) = my(d=digits(n, b)); d == Vecrev(d);
a(n) = my(b=2, N=2^n); while (! ispal(N, b), b++); b;

a(n) = {my(pow2 = 1<David A. Corneth, Jan 18 2024

from itertools import count
from sympy.ntheory.factor_ import digits
def A369233(n):
    m = 1<>1)]==s[:-t-1:-1]) # Chai Wah Wu, Jan 17 2024

a(n) = A016026(A000079(n)).

A372754 a(n) is the least base in which the Fibonacci number A000045(n) is a palindrome.

Original entry on oeis.org

2, 2, 3, 2, 2, 3, 3, 2, 4, 4, 8, 11, 3, 11, 9, 11, 15, 29, 25, 29, 29, 29, 40, 17, 121, 76, 76, 69, 147, 39, 148, 199, 199, 199, 311, 361, 10876, 428, 521, 521, 1026, 1364, 1025, 1364, 1364, 1364, 2100, 2018, 4973, 3571, 3571, 3571, 5802, 6461, 11343, 9349, 9349, 9349, 31952, 24476, 15885, 24476

Offset: 1

Robert Israel, May 12 2024

With F = A000045 the Fibonacci numbers and L = A000032 the Lucas numbers, for j odd we have F(3*j+k) = F(j+k)*L(j)^2 + F(k)*L(j) + F(j+k), thus this is a palindrome mod L(j) if F(k) >= 0 and 0 <= F(j+k) < L(j). Therefore a(6*n), a(6*n+2), a(6*n+3) and a(6*n+4) all <= L(2*n+1).

			A000045(6) = 8.  In base 2 this is 1000, not a palindrome, but in base 3 it is 22, a palindrome.  Thus a(6) = 3.

Chai Wah Wu, Table of n, a(n) for n = 1..138 (terms 1..102 from Robert Israel)

Cf. A000045, A016026, A045504.

f:= proc(x) local b,L;
  for b from 2 do
    L:= convert(x,base,b);
    if L = ListTools:-Reverse(L) then return b fi
  od
end proc:
map(f, [seq(combinat:-fibonacci(n), n=1..70)]);

A372754[n_] := Block[{b = 1}, While[!PalindromeQ[IntegerDigits[#, ++b]]] & [Fibonacci[n]]; b]; Array[A372754, 70] (* Paolo Xausa, May 18 2024 *)

from itertools import count
from sympy import fibonacci
from sympy.ntheory.factor_ import digits
def A372754(n): return next(b for b in count(2) if (s := digits(fibonacci(n),b)[1:])[:(t:=len(s)+1>>1)]==s[:-t-1:-1]) # Chai Wah Wu, May 13 2024

a(n) = A016026(A000045(n)).

cp's OEIS Frontend

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

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A249634 Least number k that is a palindrome in base n but no bases less than n, or 0 if no such k exists.

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A086757 Smallest prime p such that n is a palindrome in base-p representation.

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A065437 Smallest base relative to which the n-th prime is palindromic.

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A065809 a(n) is the smallest number m > n such that m is palindromic in base n and is not palindromic in bases b with 2 <= b < n.

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A369233 Smallest base for which the digits expansion of 2^n is palindromic.

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A372754 a(n) is the least base in which the Fibonacci number A000045(n) is a palindrome.

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