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A034677 Sum of cubes of unitary divisors of n. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified November 23 21:51 EST 2020. Contains 338603 sequences. (Running on oeis4.)