A214424 Numbers that are palindromic in exactly two bases b, 2 <= b <= 10.

15, 16, 17, 18, 20, 24, 26, 27, 28, 31, 33, 36, 45, 46, 50, 51, 52, 57, 67, 73, 78, 82, 88, 91, 92, 93, 98, 99, 104, 105, 107, 109, 111, 114, 119, 127, 129, 135, 141, 142, 150, 151, 160, 170, 171, 173, 182, 185, 186, 200, 209, 212, 215, 219, 227, 246, 252

Offset: 1

Author

T. D. Noe, Jul 18 2012

Keywords

• nonn
• base

Comments

Every pair of bases occurs. The pair (2,3), for the number a(732) = 1422773, is the last to occur. Note that 1422773 = 101011011010110110101(2) = 2200021200022(3).

See A238338 for the pairs of bases. - T. D. Noe, Mar 07 2014

Examples

			15 is palindromic in bases 2 and 4: 15 = 1111_2 = 33_4.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
• Attila Bérczes and Volker Ziegler, On simultaneous palindromes, arXiv 1403.0787 [math.NT], 2014.
• Edray Herber Goins, Palindromes in different bases: a conjecture of J. Ernest Wilkins, Integers, Vol. 9 (2009), A55.

Crossrefs

Cf. A050813, A214423, A214425, A214426 (palindromic in 0-1 and 3-4 bases).

Programs

Mathematica

n = -1; t = {}; While[Length[t] < 100, n++; If[Count[Table[s = IntegerDigits[n, m]; s == Reverse[s], {m, 2, 10}], True] == 2, AppendTo[t, n]]]; t

PARI

pal(v)=v==Vecrev(v)
is(n)=sum(b=2,10,pal(digits(n,b)))==2 \\ Charles R Greathouse IV, Mar 05 2014

Formula

A050812(a(n)) = 2.