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

Author

Erich Friedman

Keywords

• nonn
• mult

Comments

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

Examples

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

Links

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

Crossrefs

Cf. A034444, A034448.

Row n=3 of A286880.

Programs

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 *)

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 */

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