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A034676 Sum of squares of unitary divisors of n.

%I A034676 #33 Feb 16 2025 08:32:37
%S A034676 1,5,10,17,26,50,50,65,82,130,122,170,170,250,260,257,290,410,362,442,
%T A034676 500,610,530,650,626,850,730,850,842,1300,962,1025,1220,1450,1300,
%U A034676 1394,1370,1810,1700,1690,1682,2500,1850,2074,2132,2650,2210,2570,2402,3130
%N A034676 Sum of squares of unitary divisors of n.
%C A034676 Also sum of unitary divisors of n^2. - _Vladeta Jovovic_, Nov 13 2001
%C A034676 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 A034676 Reinhard Zumkeller, <a href="/A034676/b034676.txt">Table of n, a(n) for n = 1..10000</a>
%H A034676 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnitaryDivisorFunction.html">Unitary Divisor Function</a>.
%H A034676 Wikipedia, <a href="http://en.wikipedia.org/wiki/Unitary_divisor">Unitary divisor</a>.
%F A034676 Multiplicative with a(p^e)=p^(2*e)+1.
%F A034676 Dirichlet g.f.: zeta(s)*zeta(s-2)/zeta(2*s-2). - _R. J. Mathar_, Mar 04 2011
%F A034676 Sum_{k=1..n} a(k) ~ 30 * Zeta(3) * n^3 / Pi^4. - _Vaclav Kotesovec_, Jan 11 2019
%F A034676 Sum_{k>=1} 1/a(k) = 1.5594563610641446770272272038182777336348840179730233519185104374159616326... - _Vaclav Kotesovec_, Sep 20 2020
%p A034676 A034676 := proc(n)
%p A034676     a :=1 ;
%p A034676     for pe in ifactors(n)[2] do
%p A034676         p := op(1,pe) ;
%p A034676         e := op(2,pe) ;
%p A034676         a := a*(p^(2*e)+1) ;
%p A034676     end do:
%p A034676     a ;
%p A034676 end proc:
%p A034676 seq(A034676(n),n=1..40) ; # _R. J. Mathar_, Jul 12 2024
%t A034676 f[n_] := Block[{d = Divisors@ n}, Plus @@ (Select[d, GCD[#, n/#] == 1 &]^2)]; Array[f, 50] (* _Robert G. Wilson v_, Mar 04 2011 *)
%t A034676 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 A034676 (PARI) A034676_vec(len)={
%o A034676         a000012=direuler(p=2,len, 1/(1-X)) ;
%o A034676         a000290=direuler(p=2,len, 1/(1-p^2*X)) ;
%o A034676         a000290x=direuler(p=2,len, 1-p^2*X^2) ;
%o A034676         dirmul(dirmul(a000012,a000290),a000290x)
%o A034676 }
%o A034676 A034676_vec(70) ; /* via D.g.f., _R. J. Mathar_, Mar 05 2011 */
%o A034676 (Haskell)
%o A034676 a034676 = sum . map (^ 2) . a077610_row
%o A034676 -- _Reinhard Zumkeller_, Feb 12 2012
%Y A034676 Cf. A034444, A034448, A034677, A034678-A034682.
%Y A034676 Cf. A077610.
%K A034676 nonn,mult
%O A034676 1,2
%A A034676 _Erich Friedman_