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
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 *)
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
KEYWORD
nonn,mult
AUTHOR
STATUS
approved