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A016026 Smallest base relative to which n is palindromic. 5
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

Vincenzo Librandi and 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)

A016026(A016038(n)) = A016038(n) - 1, for n > 3.

A016026(A006995(n)) = 2, for n > 1.

A016026(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

CROSSREFS

Cf. A016038, A006995, A002113.

Sequence in context: A112047 A112048 A060395 * A086757 A166985 A046215

Adjacent sequences:  A016023 A016024 A016025 * A016027 A016028 A016029

KEYWORD

nonn,base

AUTHOR

Robert G. Wilson v

STATUS

approved

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Last modified February 27 15:11 EST 2020. Contains 332307 sequences. (Running on oeis4.)