A215244 -id:A215244 - cp's OEIS frontend

A220654 Smallest number that can be written in binary representation in exactly n ways as concatenation of palindromes.

0, 3, 9, 7, 17, 23, 34, 15, 33, 39, 55, 47, 66, 88, 72, 31, 65, 71, 103, 79, 133, 111, 152, 95, 130, 136, 215, 176, 277, 144, 273, 63, 129, 135, 199, 143, 443, 207, 284, 159, 261, 280, 239, 223, 588, 260, 264, 191, 258, 376, 272, 336, 627, 431, 529, 352, 532

Offset: 1

Reinhard Zumkeller, Dec 17 2012

A215244(a(n)) = n and A215244(m) < n for m < a(n).

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a220654 = fromJust . (`elemIndex` a215244_list)

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.

Original entry on oeis.org

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

Rémy Sigrist, Jan 17 2020

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

			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.

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

Cf. A000120, A006995, A215244, A331362.

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
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See Links section.
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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.

A330720 a(n) is the number of ways of writing the binary expansion of n as a product (or concatenation) of nonpalindromes.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 2, 1, 1, 1, 1, 0, 1, 2, 2, 2, 2, 0, 2, 1, 1, 2, 2, 0, 1, 1, 1, 0, 1, 3, 3, 3, 3, 3, 3, 2, 2, 3, 4, 2, 2, 2, 3, 1, 1, 3, 2, 1, 2, 2, 3, 1, 1, 2, 2, 1, 1, 1, 1, 0, 1, 4, 4, 4, 3, 5, 5, 3, 3, 4, 4, 4, 4, 5, 5, 2, 2, 5, 4, 4, 4, 0, 4

Offset: 0

Rémy Sigrist, Dec 28 2019

This sequence is a variant of A215244.

			For n = 41:
- the binary expansion of 41 is "101001",
- the possible products of nonpalindromes are "101001", "1010"."01", and "10"."10"."01",
- hence a(41) = 3.

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

Cf. A020988, A154809, A215244, A301453.

ispali:= proc(L) L = ListTools:-Reverse(L) end proc:
g:= proc(L) option remember; local m;
    add(procname(L[m+1..-1]), m= remove(t -> ispali(L[1..t]),[$1..nops(L)]))
end proc:
g([]):= 1:
seq(g(convert(n,base,2)),n=0..100); # Robert Israel, Dec 29 2019

a(n) = my (b=binary(n), v=b!=Vecrev(b)); for (s=1, #b, my (z=b[s..#b]); if (z!=Vecrev(z), v+=a(fromdigits(b[1..s-1],2)))); v

a(2^k-1) = 0 for any k >= 0.

a(A020988(k+1)) = 2^k for any k >= 0.

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A220654 Smallest number that can be written in binary representation in exactly n ways as concatenation of palindromes.

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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.

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A330720 a(n) is the number of ways of writing the binary expansion of n as a product (or concatenation) of nonpalindromes.

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