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A091627 Number of ordered integer pairs (b,c) with 1 <= b,c <= n such that both roots of x^2+bx+c=0 are integers.

%I A091627 #36 Feb 16 2025 08:32:52
%S A091627 0,0,1,2,4,5,7,8,10,12,14,15,18,19,21,23,26,27,30,31,34,36,38,39,43,
%T A091627 45,47,49,52,53,57,58,61,63,65,67,72,73,75,77,81,82,86,87,90,93,95,96,
%U A091627 101,103,106,108,111,112,116,118,122,124,126,127,133,134,136,139,143
%N A091627 Number of ordered integer pairs (b,c) with 1 <= b,c <= n such that both roots of x^2+bx+c=0 are integers.
%C A091627 Also number of ordered pairs of positive integers (i, j) such that i+j <= n and i*j <= n. - _Seiichi Manyama_, Sep 04 2021
%H A091627 Seiichi Manyama, <a href="/A091627/b091627.txt">Table of n, a(n) for n = 0..10000</a>
%H A091627 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuadraticEquation.html">Quadratic Equation</a>
%F A091627 a(n) = A091626(n) - n - 1. a(n) = a(n-1) + ceiling(tau(n)/2), n>1. Partial sums of A038548. - _Vladeta Jovovic_, Jun 12 2004
%F A091627 G.f.: (1/(1 - x)) * (-x + Sum_{k>=1} x^(k^2)/(1 - x^k)). - _Seiichi Manyama_, Sep 04 2021
%t A091627 Accumulate[ Join[{0, 0}, Table[ Ceiling[ DivisorSigma[0, n]/2], {n, 2, 64}]]]  (* _Jean-François Alcover_, Oct 23 2012, after _Vladeta Jovovic_ *)
%o A091627 (PARI) a(n) = sum(i=1, n, sum(j=i, n-i, i*j<=n)); \\ _Seiichi Manyama_, Sep 04 2021
%o A091627 (PARI) N=66; x='x+O('x^N); concat([0, 0], Vec((-x+sum(k=1, sqrtint(N), x^k^2/(1-x^k)))/(1-x))) \\ _Seiichi Manyama_, Sep 04 2021
%o A091627 (Python)
%o A091627 from math import isqrt
%o A091627 def A091627(n):
%o A091627     m = isqrt(n)
%o A091627     return 0 if n == 0 else sum(n//k for k in range(1, m+1))-m*(m-1)//2-1 # _Chai Wah Wu_, Oct 07 2021
%Y A091627 Cf. A000005, A067274, A091626, A094820.
%K A091627 nonn
%O A091627 0,4
%A A091627 _Eric W. Weisstein_, Jan 24 2004