cp's OEIS Frontend

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.

%I A100251 #11 Mar 12 2023 14:15:47
%S A100251 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,
%T A100251 9,35,451,21,33,69,14,124,6,44,715,28,24,7421,217,34,16,23001,54,1065,
%U A100251 36,7,76,156,0,245
%N 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.
%C A100251 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.
%F A100251 a(n)^2 = 1 + (1+1*n) + (1+2*n) + ... + (1+A100254(n)*n) = 1 + (1+1*n) +(1+2*n) + ... + A100253(n).
%F A100251 a(n)^2 = A100252(n).
%e A100251 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.
%t A100251 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 *)
%Y A100251 Cf. A100252, A100253.
%K A100251 nonn
%O A100251 3,1
%A A100251 _Charlie Marion_, Nov 21 2004