A123586 -id:A123586 - cp's OEIS frontend

A060873 Intrinsic 3-palindromes: n is an intrinsic k-palindrome if it is a k-digit palindrome in some base.

5, 7, 10, 13, 16, 17, 20, 21, 23, 25, 26, 29, 31, 34, 36, 37, 38, 41, 42, 43, 46, 49, 50, 51, 52, 55, 57, 59, 61, 62, 63, 64, 65, 67, 71, 72, 73, 74, 78, 80, 81, 82, 83, 85, 86, 88, 89, 91, 92, 93, 97, 98, 100, 101, 104, 105, 107, 109, 111, 113, 114, 117, 118

Offset: 1

Harvey P. Dale, May 05 2001

All numbers are intrinsic 1- and (except 1 and 2) 2-palindromes, almost all numbers are intrinsic 3-palindromes and very few numbers are intrinsic k-palindromes for k >= 4.

Jean-François Alcover, Table of n, a(n) for n = 1..1000
A. J. Di Scala and M. Sombra, Intrinsic Palindromic Numbers, arXiv:math/0105022 [math.GM], 2001.
A. J. Di Scala and M. Sombra, Intrinsic Palindromes, Fib. Quart. 42, no. 1, Feb. 2004, pp. 76-81.

Cf. A060874-A060879, A060947-A060949, A123586, A114255.

testQ[n_, k_] := For[b = 2, b <= Ceiling[(n-1)^(1/(k-1))], b++, d = IntegerDigits[n, b]; If[Length[d] == k && d == Reverse[d], Return[True]]]; n0[k_] := 2^(k-1) + 1; Reap[Do[If[testQ[n, 3] === True, Print[n, " ", FromDigits[d], " b = ", b]; Sow[n]], {n, n0[3], 200}]][[2, 1]] (* Jean-François Alcover, Nov 07 2014 *)

A114255 Numbers that are nontrivial (3 digits or more) palindromes when expressed in some base 2 or greater.

Original entry on oeis.org

5, 7, 9, 10, 13, 15, 16, 17, 20, 21, 23, 25, 26, 27, 28, 29, 31, 33, 34, 36, 37, 38, 40, 41, 42, 43, 45, 46, 49, 50, 51, 52, 55, 56, 57, 59, 61, 62, 63, 64, 65, 67, 68, 71, 72, 73, 74, 78, 80, 81, 82, 83, 85, 86, 88, 89, 91, 92, 93, 97, 98, 99, 100, 101, 104, 105, 107, 109

Offset: 1

Jason Orendorff (jason.orendorff(AT)gmail.com), Feb 05 2006

All integers are trivially palindromes in base 1. All integers n>2 are trivially 2-digit palindromes because they can be represented as "11" in base n-1.

			5 is present because the palindrome (101 base 2) = 5; 803 is present because (30203 base 4) = 803.

Cf. A060873-A060879, A060947-A060949, A123586.

isPalindrome s = (s == reverse s) digits 0 _ = [] digits n b = n `rem` b : digits (n `quot` b) b check n = any isPalindrome $ takeWhile (\x -> length x > 2) $ map (digits n) [2..] main = mapM print $ filter check [1..]

palindromeQ[n_, b_] := (id = IntegerDigits[n, b]) === Reverse[id] && Length[id] >= 3; palindromeQ[n_] := Or @@ (palindromeQ[n, #] & ) /@ Range[2, n-2]; Select[ Range[110], palindromeQ] (* Jean-François Alcover, Dec 16 2011 *)

isok(n) = for (b=2, n-1, if ((d=digits(n,b)) && (#d >= 3) && (Vecrev(d) == d), return (1));); \\ Michel Marcus, Jul 28 2016

Cross-references from Charles R Greathouse IV, Aug 04 2010

A100563 Number of bases less than sqrt(n) in which n is a palindrome.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 2, 0, 0, 1, 2, 0, 1, 0, 1, 2, 1, 1, 1, 0, 2, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 0, 0, 1, 1, 2, 2, 0, 0, 2, 1, 2, 0, 1, 0, 1, 1, 2, 1, 3, 0, 2, 1, 0, 0, 1, 1, 2, 1, 0, 0, 0, 2, 0, 2, 1, 2, 1, 0, 3, 1, 0, 1, 1, 0, 2, 2, 2, 0, 0, 0, 1, 2, 1, 3, 1, 0, 0, 2, 2

Offset: 1

Gordon Hamilton, Nov 29 2004

Is there a number m such that a(n) > 0 for all n > m? I call the set of numbers for which a(n)=0 "unkempt" for refusing to use a mirror in any base. Is there an infinite number of unkempt numbers? a(n) can be arbitrarily large.
The sequence A123586 gives the values of n where a(n)=0. - Robert G. Wilson v, Nov 01 2014
Is there a closed-form formula for this function? - Robert G. Wilson v, Nov 01 2014
From Robert G. Wilson v, Nov 26 2014: (Start)
The first occurrence, beginning at 0, of n is: 1, 5, 17, 65, 121, 562, 1432, 1477, 4369, 36582, 35101, 86677, 83161, 360361, 291721, 720721, 887041, 1496881, 1670761, 3931201, 3341521, 5654881, 7207201, 7761601,...
Positions where a(n)=k:
k = 0:  A123586;
k = 1:  5, 7, 9, 10, 13, 15, 16, 20, 23, 25, 27, 28, 29, 33, 34, 36, 37, 38, 40, ...;
k = 2:  17, 21, 26, 31, 46, 51, 52, 55, 57, 63, 67, 73, 78, 80, 82, 91, 92, 93, 98, ...;
k = 3:  65, 85, 100, 130, 154, 164, 170, 178, 191, 195, 203, 209, 242, 282, 292, ...;
k = 4:  121, 235, 255, 257, 273, 300, 325, 341, 343, 373, 400, 495, 601, 610, 626, 666, ...;
k = 5:  562, 676, 771, 819, 1009, 1111, 1220, 1333, 1365, 1441, 1543, 1978, 1981, 2000, ...;
k = 6:  1432, 2380, 2666, 2925, 3280, 4035, 4095, 4161, 4225, 4401, 4525, 4561, 4681, ...;
k = 7:  1477, 4097, 4591, 7141, 7993, 8191, 9640, 10081, 10297, 10626, 10858, 11761, ...; etc.
(End)

			100 is a palindrome in bases 3, 7 and 9, so a(100) = 3.

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

Cf. A060873, A060874, A060875, A060876, A060877, A060878, A060879, A060947, A060948, A060949, A123586.

Cf. A000042.

f[n_] := Module[{p}, Table[ p = IntegerDigits[n, b]; If[p == Reverse@ p, {b, p}, Sequence @@ {}], {b, 2, Sqrt@ n}]]; Array[ Length@ f@# &, 105] (* Robert G. Wilson v, Nov 01 2014 *)

a(n) = {my(nb = 0); for (b=2, sqrt(n), d = digits(n, b); nb+= (Vecrev(d) == d);); nb;} \\ Michel Marcus, Nov 05 2014

a(n) = A135551(n) - A033831(n). - Robert G. Wilson v, Nov 01 2014

a(58) from Robert G. Wilson v, Nov 05 2014

A275220 a(n) is the smallest integer with at least n palindromic representations of length >= 3 in bases b >= 2.

Original entry on oeis.org

5, 17, 65, 121, 562, 1432, 1477, 4369, 35101, 35101, 83161, 83161, 291721, 291721, 720721, 887041, 1496881, 1670761, 3341521, 3341521, 5654881, 7207201, 7761601, 10810801, 12252241, 15135121, 21621601, 28274401, 28274401, 41081041, 41081041, 41081041

Offset: 1

Clive Stott, Jul 20 2016

a(9) and a(10) are identical because there is no number less than 35101 with nine palindromic representations in different bases; likewise for a(11) and a(12), and for a(13) and a(14).

Cf. A123586.

isok(k, n) = sum(b=2, k-1, (d=digits(k,b)) && (#d >= 3) && (Vecrev(d) == d)) >= n;
a(n) = my(j=1); while(! isok(j, n), j++); j; \\ Michel Marcus, Jul 28 2016

a(17)-a(32) from Giovanni Resta, Jul 28 2016

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A060873 Intrinsic 3-palindromes: n is an intrinsic k-palindrome if it is a k-digit palindrome in some base.

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A114255 Numbers that are nontrivial (3 digits or more) palindromes when expressed in some base 2 or greater.

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A100563 Number of bases less than sqrt(n) in which n is a palindrome.

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A275220 a(n) is the smallest integer with at least n palindromic representations of length >= 3 in bases b >= 2.

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