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A091626
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Number of ordered integer pairs (b,c) with 0 <= b, c <= n such that both roots of x^2+bx+c=0 are integers.
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4
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1, 2, 4, 6, 9, 11, 14, 16, 19, 22, 25, 27, 31, 33, 36, 39, 43, 45, 49, 51, 55, 58, 61, 63, 68, 71, 74, 77, 81, 83, 88, 90, 94, 97, 100, 103, 109, 111, 114, 117, 122, 124, 129, 131, 135, 139, 142, 144, 150, 153, 157, 160, 164, 166, 171, 174, 179, 182, 185, 187
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OFFSET
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0,2
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COMMENTS
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Also number of ordered pairs of nonnegative integers (i, j) such that i+j <= n and i*j <= n. - Seiichi Manyama, Sep 04 2021
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LINKS
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Griffin N. Macris, Table of n, a(n) for n = 0..9999
Eric Weisstein's World of Mathematics, Quadratic Equation
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FORMULA
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a(n) = a(n-1) + ceiling(tau(n)/2) + 1, n>1. - Vladeta Jovovic, Jun 12 2004
a(n) = n + floor(sqrt(n))/2 + A006218(n)/2, n>0. - Griffin N. Macris, Jun 14 2016
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EXAMPLE
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The six quadratics for a(3)=6 and their roots are as follows:
x^2 + 0*x + 0; x=0.
x^2 + 1*x + 0; x=0, x=-1.
x^2 + 2*x + 0; x=0, x=-2.
x^2 + 2*x + 1; x=-1.
x^2 + 3*x + 0; x=0, x=-3.
x^2 + 3*x + 2; x=-1, x=-2.
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MATHEMATICA
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a[n_] := a[n] = a[n-1] + Ceiling[ DivisorSigma[0, n]/2] + 1; a[0]=1; a[1]=2; Table[a[n], {n, 0, 59}] (* Jean-François Alcover, Nov 08 2012, after Vladeta Jovovic *)
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PROG
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(PARI) a(n) = sum(i=0, n, sum(j=i, n-i, i*j<=n)); \\ Seiichi Manyama, Sep 04 2021
(Python)
from math import isqrt
def A091626(n):
m = isqrt(n)
return 1 if n == 0 else n+sum(n//k for k in range(1, m+1))-m*(m-1)//2 # Chai Wah Wu, Oct 07 2021
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CROSSREFS
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Cf. A000005, A006218, A038548, A067274, A091627.
Sequence in context: A265286 A186343 A224995 * A085223 A163057 A242137
Adjacent sequences: A091623 A091624 A091625 * A091627 A091628 A091629
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KEYWORD
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nonn
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AUTHOR
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Eric W. Weisstein, Jan 24 2004
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STATUS
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approved
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