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

1,2

Links

Table of n, a(n) for n=1..61.

László Tóth, Multiplicative arithmetic functions of several variables: a survey, arXiv preprint arXiv:1310.7053 [math.NT], 2013.

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)

Mathematica

a[n_] := n * DivisorSum[n, DivisorSigma[0, #]^3/# &]; Array[a, 61] (* Amiram Eldar, May 09 2021 *)

Prog

(PARI) a(n) = n*sumdiv(n, d, numdiv(d)^3/d);
(PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, numdiv(k)^3*x^k/(1-x^k)^2))

Crossrefs

Cf. A007429, A013661, A062369, A097988, A344043.

Sequence in context: A042395 A041210 A267340 * A042413 A041212 A257313

Adjacent sequences: A344079 A344080 A344081 * A344083 A344084 A344085

Keyword

nonn,mult,easy

Author

Seiichi Manyama, May 09 2021

Status

approved