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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. 9

%I #35 Oct 07 2021 20:59:36

%S 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 45,47,49,52,53,57,58,61,63,65,67,72,73,75,77,81,82,86,87,90,93,95,96,

%U 101,103,106,108,111,112,116,118,122,124,126,127,133,134,136,139,143

%N 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 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 Seiichi Manyama, <a href="/A091627/b091627.txt">Table of n, a(n) for n = 0..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/QuadraticEquation.html">Quadratic Equation</a>

%F 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 G.f.: (1/(1 - x)) * (-x + Sum_{k>=1} x^(k^2)/(1 - x^k)). - _Seiichi Manyama_, Sep 04 2021

%t Accumulate[ Join[{0, 0}, Table[ Ceiling[ DivisorSigma[0, n]/2], {n, 2, 64}]]] (* _Jean-François Alcover_, Oct 23 2012, after _Vladeta Jovovic_ *)

%o (PARI) a(n) = sum(i=1, n, sum(j=i, n-i, i*j<=n)); \\ _Seiichi Manyama_, Sep 04 2021

%o (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 (Python)

%o from math import isqrt

%o def A091627(n):

%o m = isqrt(n)

%o 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 Cf. A000005, A067274, A091626, A094820.

%K nonn

%O 0,4

%A _Eric W. Weisstein_, Jan 24 2004

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Last modified August 13 04:06 EDT 2024. Contains 375113 sequences. (Running on oeis4.)