A065610 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.

1, 47, 2, 5, 767, 16, 1727, 22, 17, 13, 18, 112, 10, 70, 8, 10799, 12287, 21, 82, 17327, 31, 15, 255, 16, 10, 13, 9, 5, 49, 40367, 43199, 117, 17, 1630, 7, 58799, 10, 65711, 34, 73007, 49, 13, 64, 29, 17, 6, 9, 30, 42, 309, 8, 124847, 17, 31, 139967, 13, 150527, 15

Offset: 0

Author

Labos Elemer, Nov 07 2001

Keywords

• nonn

Comments

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.

Examples

			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).
		

Crossrefs

Cf. A000330, A065311, A065312, A065313, A065314, A065315.

Programs

Mathematica

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 *)

Formula

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.