A331362 a(n) is the greatest value of the form s_1 + ... + s_k such that the concatenation of the binary representations of s_1^2, ..., s_k^2 equals the binary representation of n.

0, 1, 1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 3, 3, 4, 4, 3, 3, 4, 3, 3, 3, 4, 3, 5, 3, 4, 4, 4, 4, 5, 4, 5, 3, 4, 6, 4, 4, 5, 3, 4, 3, 4, 4, 4, 4, 5, 5, 7, 5, 6, 4, 4, 4, 5, 4, 6, 4, 5, 5, 5, 5, 6, 8, 5, 5, 6, 4, 4, 4, 5, 6, 7, 4, 5, 5, 5, 5, 6, 5, 9, 4, 5, 4, 4, 4

Offset

0,4

Comments

As 0 and 1 are squares, we can always split the binary representation of a number into squares, and the sequence is well defined.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..16384

Rémy Sigrist, Density plot of the first 2^20 terms

Rémy Sigrist, PARI program for A331362

Index entries for sequences related to binary expansion of n

Formula

a(n) >= A000120(n) with equality iff n belongs to A003754.

a(n^2) = n.

Example

For n = 8:
- the binary representation of 8 is "1000",
- we can split it into "100" and "0" (2^2 and 0^2),
- or into "1" and "0" and "0" and "0" (1^2 and 0^2 and 0^2 and 0^2),
- so a(8) = max(2+0, 1+0+0+0) = 2.

Prog

(PARI) See Links section.

Crossrefs

Cf. A000120, A003754.

Sequence in context: A260235 A078120 A057525 * A139325 A341829 A344888

Adjacent sequences: A331359 A331360 A331361 * A331363 A331364 A331365

Keyword

nonn,base

Author

Rémy Sigrist, Jan 14 2020

Status

approved