A034676 - OEIS

%I #32 Jul 12 2024 10:12:05

%S 1,5,10,17,26,50,50,65,82,130,122,170,170,250,260,257,290,410,362,442,

%T 500,610,530,650,626,850,730,850,842,1300,962,1025,1220,1450,1300,

%U 1394,1370,1810,1700,1690,1682,2500,1850,2074,2132,2650,2210,2570,2402,3130

%N Sum of squares of unitary divisors of n.

%C Also sum of unitary divisors of n^2. - _Vladeta Jovovic_, Nov 13 2001

%C 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

%H Reinhard Zumkeller, <a href="/A034676/b034676.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UnitaryDivisorFunction.html">Unitary Divisor Function</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Unitary_divisor">Unitary divisor</a>.

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

%F Dirichlet g.f.: zeta(s)*zeta(s-2)/zeta(2*s-2). - _R. J. Mathar_, Mar 04 2011

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

%F Sum_{k>=1} 1/a(k) = 1.5594563610641446770272272038182777336348840179730233519185104374159616326... - _Vaclav Kotesovec_, Sep 20 2020

%p A034676 := proc(n)

%p     a :=1 ;

%p     for pe in ifactors(n)[2] do

%p         p := op(1,pe) ;

%p         e := op(2,pe) ;

%p         a := a*(p^(2*e)+1) ;

%p     end do:

%p     a ;

%p end proc:

%p seq(A034676(n),n=1..40) ; # _R. J. Mathar_, Jul 12 2024

%t f[n_] := Block[{d = Divisors@ n}, Plus @@ (Select[d, GCD[#, n/#] == 1 &]^2)]; Array[f, 50] (* _Robert G. Wilson v_, Mar 04 2011 *)

%t f[p_, e_] := p^(2*e)+1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 14 2020 *)

%o (PARI) A034676_vec(len)={

%o         a000012=direuler(p=2,len, 1/(1-X)) ;

%o         a000290=direuler(p=2,len, 1/(1-p^2*X)) ;

%o         a000290x=direuler(p=2,len, 1-p^2*X^2) ;

%o         dirmul(dirmul(a000012,a000290),a000290x)

%o }

%o A034676_vec(70) ; /* via D.g.f., _R. J. Mathar_, Mar 05 2011 */

%o (Haskell)

%o a034676 = sum . map (^ 2) . a077610_row

%o -- _Reinhard Zumkeller_, Feb 12 2012

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

%Y Cf. A077610.

%K nonn,mult

%O 1,2

%A _Erich Friedman_