A265026 First differences of A048701.

3, 6, 6, 18, 12, 6, 12, 66, 24, 12, 24, 6, 24, 12, 24, 258, 48, 24, 48, 12, 48, 24, 48, 6, 48, 24, 48, 12, 48, 24, 48, 1026, 96, 48, 96, 24, 96, 48, 96, 12, 96, 48, 96, 24, 96, 48, 96, 6, 96, 48, 96, 24, 96, 48, 96, 12, 96, 48, 96, 24, 96, 48

Offset: 1

Author

N. J. A. Sloane, Nov 30 2015

Keywords

• nonn

Comments

Comment from Altug Alkan, Dec 03 2015: (Start) Except for 3, all terms are divisible by 6 (cf. A048702, A265027).

Proof: Binary palindromes of even length (A048701) are odd for n > 0. So A048701(n) - A048701(n-1) is an even number for n > 1. Because the length is even and palindromic numbers are symmetric, for any digit “1” that is related with 2^n in its expansion which n is even, there are another digit “1” that is related with 2^m in its expansion which m is odd. 2^n+2^m is always divisible by 3 if n is even and m is odd. Therefore A048701(n) is divisible by 3, so A048701(n) - A048701(n-1) is divisible by 3 for n > 0. In conclusion, A048701(n) - A048701(n-1) is always divisible by 6 for n > 1. (End)

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..32767

Crossrefs

Cf. A048701, A048702, A265027.

Programs

PARI

a048701(n) = my(f); f = length(binary(n)) - 1; 2^(f+1)*n + sum(i=0, f, bittest(n, i) * 2^(f-i));
vector(100, n, (a048701(n) - a048701(n-1))) \\ Altug Alkan, Dec 03 2015

Formula

a(n) = A048701(n) - A048701(n-1). - Altug Alkan, Dec 03 2015