A331471 Consider the different ways to split the binary representation of n into palindromic parts; a(n) is the greatest possible sum of the parts of such a split.

0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 6, 3, 6, 7, 15, 1, 17, 9, 10, 5, 21, 7, 8, 3, 10, 6, 27, 7, 8, 15, 31, 1, 33, 17, 18, 9, 10, 10, 12, 5, 10, 21, 22, 7, 45, 15, 16, 3, 18, 10, 51, 6, 22, 27, 28, 7, 12, 9, 28, 15, 16, 31, 63, 1, 65, 33, 34, 17, 18, 18, 20, 9, 73

Offset: 0

Author

Rémy Sigrist, Jan 17 2020

Keywords

• nonn
• base

Comments

Leading zeros are forbidden in the binary representation of n; however we allow leading zeros in the palindromic parts.

Examples

			For n = 10:
- the binary representation of 10 is "1010",
- we can split it into "1" and "0" and "1" and "0" (1 and 0 and 1 and 0),
- or into "101" and "0" (5 and 0),
- or into "1" and "010" (1 and 2),
- hence a(n) = max(2, 5, 3) = 5.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..8192
• Rémy Sigrist, PARI program for A331471
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A000120, A006995, A215244, A331362.

Programs

Mathematica

palQ[w_] := w == Reverse@w; ric[tg_, cr_] := Block[{m = Length@tg, t}, If[m == 0, Sow@ Total[ FromDigits[#, 2] & /@ cr], Do[ If[ palQ[t = Take[tg, k]], ric[Drop[tg, k], Join[cr, {t}]]], {k, m}]]]; a[n_] := Max[ Reap[ ric[ IntegerDigits[n, 2], {}]][[2, 1]]]; a /@ Range[0, 73] (* Giovanni Resta, Jan 19 2020 *)

PARI

See Links section.

Formula

a(n) >= A000120(n) with equality iff n = 0 or n is a power of 2.

a(n) <= n with equality iff n belongs to A006995.