A100251 The square root of A100252; the index of the least square number greater than 1 that is also an n-gonal number, or 0 if none exists.

6, 2, 99, 35, 9, 15, 3, 0, 14, 8, 6, 21, 55, 4, 133, 10, 22, 0, 51, 27, 261, 15, 5, 85, 161, 9, 35, 451, 21, 33, 69, 14, 124, 6, 44, 715, 28, 24, 7421, 217, 34, 16, 23001, 54, 1065, 36, 7, 76, 156, 0, 245

Offset: 3

Author

Charlie Marion, Nov 21 2004

Keywords

• nonn

Comments

Let j be the smallest integer for which 1 + (1+1*n) + (1+2*n) + ... + (1+j*n) = k^2 = s. Then a(n)=k; if no such j exists, then a(n)=0. Basis for sequence is shortest arithmetic series with initial term 1 and difference n that sums to a perfect square.

Examples

			a(3)=99 since 1 + 4 + 7 + ... + (1+80*3) = 99^2 = 9801 and no other arithmetic series with initial term 1, difference 3 and fewer terms sums to a perfect square.
		

Crossrefs

Cf. A100252, A100253.

Programs

Mathematica

NgonIndex[n_, v_] := (-4 + n + Sqrt[16 - 8*n + n^2 - 16*v + 8*n*v])/(n - 2)/2; Table[k = 2; While[sqr = k^2; i = NgonIndex[n, sqr]; k < 25000 && ! IntegerQ[i], k++]; If[k == 25000, k = sqr = i = 0]; k, {n, 3, 64}] (* T. D. Noe, Apr 19 2011 *)

Formula

a(n)^2 = 1 + (1+1*n) + (1+2*n) + ... + (1+A100254(n)*n) = 1 + (1+1*n) +(1+2*n) + ... + A100253(n).

a(n)^2 = A100252(n).