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a(n) is the greatest value of the form s_1^2 + ... + s_k^2 such that the concatenation of the binary representations of s_1^2, ..., s_k^2 equals the binary representation of n.
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%I #11 Jan 19 2020 06:27:49

%S 0,1,1,2,4,2,2,3,4,9,2,3,5,3,3,4,16,5,9,10,5,3,3,4,5,25,3,4,6,4,4,5,

%T 16,17,5,6,36,10,10,11,5,10,3,4,6,4,4,5,17,49,25,26,6,4,4,5,6,26,4,5,

%U 7,5,5,6,64,17,17,18,8,6,6,7,36,37,10,11,13,11

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

%C This sequence is a variant of A331362.

%H Rémy Sigrist, <a href="/A331470/b331470.txt">Table of n, a(n) for n = 0..8192</a>

%H Rémy Sigrist, <a href="/A331470/a331470.gp.txt">PARI program for A331470</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

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

%F a(n^2) = n^2.

%e For n = 12:

%e - the binary representation of 12 is "1100",

%e - we can split it into "1" and "1" and "0" and "0" (1^2 and 1^2 and 0^2 and 0^2),

%e - or into "1" and "100" (1^2 and 2^2),

%e - hence a(12) = max(2, 5) = 5.

%o (PARI) See Links section.

%Y Cf. A000120, A003754, A331362.

%K nonn,base

%O 0,4

%A _Rémy Sigrist_, Jan 17 2020