A344082 a(n) = n * Sum_{d|n} tau(d)^3 / d, where tau(n) is the number of divisors of n.
1, 10, 11, 47, 13, 110, 15, 158, 60, 130, 19, 517, 21, 150, 143, 441, 25, 600, 27, 611, 165, 190, 31, 1738, 92, 210, 244, 705, 37, 1430, 39, 1098, 209, 250, 195, 2820, 45, 270, 231, 2054, 49, 1650, 51, 893, 780, 310, 55, 4851, 132, 920, 275, 987, 61, 2440, 247, 2370, 297, 370, 67, 6721, 69
Offset: 1
Links
- László Tóth, Multiplicative arithmetic functions of several variables: a survey, arXiv preprint arXiv:1310.7053 [math.NT], 2013.
Programs
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Mathematica
a[n_] := n * DivisorSum[n, DivisorSigma[0, #]^3/# &]; Array[a, 61] (* Amiram Eldar, May 09 2021 *)
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PARI
a(n) = n*sumdiv(n, d, numdiv(d)^3/d);
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PARI
my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, numdiv(k)^3*x^k/(1-x^k)^2))
Formula
G.f.: Sum_{k >= 1} tau(k)^3 * x^k/(1 - x^k)^2.
If p is prime, a(p) = 8 + p.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(2)^4 * Product_{p prime} (1 + 4/p^2 + 1/p^4) = 31.237542262502... . - Amiram Eldar, Dec 22 2023
From Peter Bala, Jan 25 2024: (Start)
a(n) = Sum_{d|n, e|n} gcd(d, e) * tau(n/d) * tau(n/e) (the sum is a multiplicative function of n - see Tóth).
Multiplicative: a(p^k) = ( p^(k+2)*(p^2 + 4*p + 1) - p^3*(k + 2)^3 + p^2*(3*k^3 + 15*k^2 + 21*k + 5) - p*(3*k^3 + 12*k^2 + 12*k + 4) + (k + 1)^3 ) / (p - 1)^4. (End)