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%I #13 Nov 19 2017 22:12:38
%S 1,47,2,5,767,16,1727,22,17,13,18,112,10,70,8,10799,12287,21,82,17327,
%T 31,15,255,16,10,13,9,5,49,40367,43199,117,17,1630,7,58799,10,65711,
%U 34,73007,49,13,64,29,17,6,9,30,42,309,8,124847,17,31,139967,13,150527,15
%N Smallest number m so that n^2 + A000330(m) is also a square, i.e., n^2 + (1 + 4 + 9 + 16 + ... + m^2) = w^2 for some w.
%C I.e., a(n) is the least solution to n^2 + (x(x+1)(2x+1)/6) = w^2; a(n) is the length of shortest sum of consecutive squares from 1 to a(n) which when added to n^2 gives a new square.
%F n^2 + (1 + 4 + 9 + ... + a(n)^2) = w^2, where w depends also on n; i.e., sum of consecutive squares from 1, 4, ... to a(n)^2 + n^2 is also a square.
%e n = 3: a(3) = 5 because n^2 + 1 + 4 + 9 + 16 + 25 = 9 + (1 + 4 + 9 + 16 + 25) = 64 = 8*8; n = 4: a(4) = 767 because n^2 + (1 + 4 + ... + 767^2) = 150700176 = 12276*12726, where 767 is the length of the shortest such consecutive-square sequence which provides (when summed) a new square, namely 12276^2. Often the least solution is rather large. E.g., at n = 93, a(n) = 415151, which means that 93^2 + A000330(415151) = 8649 + (long square sum) = 154436265^2 = 23850559947150225 is the smallest such square number, sum odd distinct consecutive squares except one repetition(8649).
%t s=n^2 Do[s=s+m^2; If[IntegerQ[Sqrt[s]], Print[m]], {m, 1, 500000}] (* gives solutions of which the smallest is entered into the sequence *)
%Y Cf. A000330, A065311, A065312, A065313, A065314, A065315.
%K nonn
%O 0,2
%A _Labos Elemer_, Nov 07 2001