A034677 Sum of cubes of unitary divisors of n.

1, 9, 28, 65, 126, 252, 344, 513, 730, 1134, 1332, 1820, 2198, 3096, 3528, 4097, 4914, 6570, 6860, 8190, 9632, 11988, 12168, 14364, 15626, 19782, 19684, 22360, 24390, 31752, 29792, 32769, 37296, 44226, 43344, 47450, 50654, 61740, 61544, 64638, 68922, 86688, 79508

Offset

1,2

Comments

A unitary divisor of n is a divisor d such that gcd(d,n/d)=1.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Formula

Dirichlet g.f.: zeta(s)*zeta(s-3)/zeta(2s-3). - R. J. Mathar, Mar 04 2011

If n = Product (p_j^k_j) then a(n) = Product (1 + p_j^(3*k_j)). - Ilya Gutkovskiy, Nov 04 2018

Sum_{k=1..n} a(k) ~ Pi^4 * n^4 / (360 * Zeta(5)). - Vaclav Kotesovec, Feb 01 2019

Example

The unitary divisors of 6 are 1, 2, 3 and 6, so a(6) = 252.

Mathematica

scud[n_]:=Total[Select[Divisors[n], CoprimeQ[#, n/#]&]^3]; Array[scud, 40] (* Harvey P. Dale, Oct 16 2016 *)
f[p_, e_] := p^(3*e)+1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 14 2020 *)

Prog

(PARI) A034677_vec(len)={
        a000012=direuler(p=2, len, 1/(1-X)) ;
        a000578=direuler(p=2, len, 1/(1-p^3*X)) ;
        a000578x=direuler(p=2, len, 1-p^3*X^2) ;
        dirmul(dirmul(a000012, a000578), a000578x)
}
A034677_vec(70) /* via D.g.f., R. J. Mathar, Mar 05 2011 */

Crossrefs

Cf. A034444, A034448.

Row n=3 of A286880.

Sequence in context: A321559 A041359 A034126 * A009255 A062451 A065959

Adjacent sequences: A034674 A034675 A034676 * A034678 A034679 A034680

Keyword

nonn,mult

Author

Erich Friedman

Status

approved