A034676 Sum of squares of unitary divisors of n.

1, 5, 10, 17, 26, 50, 50, 65, 82, 130, 122, 170, 170, 250, 260, 257, 290, 410, 362, 442, 500, 610, 530, 650, 626, 850, 730, 850, 842, 1300, 962, 1025, 1220, 1450, 1300, 1394, 1370, 1810, 1700, 1690, 1682, 2500, 1850, 2074, 2132, 2650, 2210, 2570, 2402, 3130

Offset

1,2

Comments

Also sum of unitary divisors of n^2. - Vladeta Jovovic, Nov 13 2001

If b(n,k)=sum of k-th powers of unitary divisors of n then b(n,k) is multiplicative with b(p^e,k)=p^(k*e)+1. - Vladeta Jovovic, Nov 13 2001

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Unitary Divisor Function.

Wikipedia, Unitary divisor.

Formula

Multiplicative with a(p^e)=p^(2*e)+1.

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

Sum_{k=1..n} a(k) ~ 30 * Zeta(3) * n^3 / Pi^4. - Vaclav Kotesovec, Jan 11 2019

Sum_{k>=1} 1/a(k) = 1.5594563610641446770272272038182777336348840179730233519185104374159616326... - Vaclav Kotesovec, Sep 20 2020

Mathematica

f[n_] := Block[{d = Divisors@ n}, Plus @@ (Select[d, GCD[#, n/#] == 1 &]^2)]; Array[f, 50] (* Robert G. Wilson v, Mar 04 2011 *)
f[p_, e_] := p^(2*e)+1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 14 2020 *)

Prog

(PARI) A034676_vec(len)={
        a000012=direuler(p=2, len, 1/(1-X)) ;
        a000290=direuler(p=2, len, 1/(1-p^2*X)) ;
        a000290x=direuler(p=2, len, 1-p^2*X^2) ;
        dirmul(dirmul(a000012, a000290), a000290x)
}
A034676_vec(70) ; /* via D.g.f., R. J. Mathar, Mar 05 2011 */
(Haskell)
a034676 = sum . map (^ 2) . a077610_row
-- Reinhard Zumkeller, Feb 12 2012

Crossrefs

Cf. A034444, A034448, A034677, A034678-A034682.

Cf. A077610.

Sequence in context: A193053 A340047 A098749 * A076598 A306011 A080341

Adjacent sequences: A034673 A034674 A034675 * A034677 A034678 A034679

Keyword

nonn,mult

Author

Erich Friedman

Status

approved