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A174466
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a(n) = Sum_{d|n} d*sigma(n/d)*tau(d).
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3
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1, 7, 10, 31, 16, 70, 22, 111, 64, 112, 34, 310, 40, 154, 160, 351, 52, 448, 58, 496, 220, 238, 70, 1110, 166, 280, 334, 682, 88, 1120, 94, 1023, 340, 364, 352, 1984, 112, 406, 400, 1776, 124, 1540, 130, 1054, 1024, 490, 142, 3510, 316, 1162, 520, 1240
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OFFSET
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1,2
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COMMENTS
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Compare to sigma_2(n) = Sum_{d|n} d*sigma(n/d)*phi(d) = sum of squares of divisors of n.
tau(n) = A000005(n) = the number of divisors of n,
and sigma(n) = A000203(n) = sum of divisors of n.
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LINKS
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FORMULA
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Dirichlet g.f. zeta(s)*(zeta(s-1))^3. - R. J. Mathar, Feb 06 2011
Sum_{k=1..n} a(k) ~ Pi^2*n^2/24 * (log(n)^2 + ((6*g - 1) + 12*z1/Pi^2) * log(n) + (1 - 6*g + 12*g^2 - 12*sg1)/2 + 6*((6*g - 1)*z1 + z2)/Pi^2), where g is the Euler-Mascheroni constant A001620, sg1 is the first Stieltjes constant A082633, z1 = Zeta'(2) = A073002, z2 = Zeta''(2) = A201994. - Vaclav Kotesovec, Feb 02 2019
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PROG
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(PARI) {a(n)=sumdiv(n, d, d*sigma(n/d)*sigma(d, 0))}
(Haskell)
a174466 n = sum $ zipWith3 (((*) .) . (*))
divs (map a000203 $ reverse divs) (map a000005 divs)
where divs = a027750_row n
(Magma) [&+[d*DivisorSigma(1, n div d)*#Divisors(d):d in Divisors(n)]:n in [1..55]]; // Marius A. Burtea, Oct 18 2019
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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STATUS
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approved
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