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

Author

Erich Friedman

Keywords

• nonn
• mult

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.

Crossrefs

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

Cf. A077610.

Programs

Haskell

a034676 = sum . map (^ 2) . a077610_row
-- Reinhard Zumkeller, Feb 12 2012

Maple

A034676 := proc(n)
    a :=1 ;
    for pe in ifactors(n)[2] do
        p := op(1,pe) ;
        e := op(2,pe) ;
        a := a*(p^(2*e)+1) ;
    end do:
    a ;
end proc:
seq(A034676(n),n=1..40) ; # R. J. Mathar, Jul 12 2024

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 *)

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 */

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