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A361794 Sum of the cubes d^3 of the divisors d satisfying d^2|n.

%I A361794 #30 Jun 10 2024 15:01:09
%S A361794 1,1,1,9,1,1,1,9,28,1,1,9,1,1,1,73,1,28,1,9,1,1,1,9,126,1,28,9,1,1,1,
%T A361794 73,1,1,1,252,1,1,1,9,1,1,1,9,28,1,1,73,344,126,1,9,1,28,1,9,1,1,1,9,
%U A361794 1,1,28,585,1,1,1,9,1,1,1,252,1,1,126,9,1,1,1,73
%N A361794 Sum of the cubes d^3 of the divisors d satisfying d^2|n.
%C A361794 The Mobius transform is 1, 0, 0, 8, 0, 0, 0, 0, 27, 0, 0, ... = n^(3/2)*A010052(n).
%H A361794 Winston de Greef, <a href="/A361794/b361794.txt">Table of n, a(n) for n = 1..10000</a>
%H A361794 A. Dixit, B. Maji, and A. Vatwani, <a href="https://arxiv.org/abs/2303.09937">Voronoi summation formula for the generalized divisor function sigma_z^k(n)</a>, arXiv:2303.09937 [math.NT], 2023, sigma(z=3,k=2,n).
%F A361794 a(n) = Sum_{d^2|n} d^3.
%F A361794 Multiplicative with a(p^e) = (p^(3*(floor(e/2) + 1)) - 1)/(p^3 - 1). - _Amiram Eldar_, Mar 24 2023
%F A361794 G.f.: Sum_{k>=1} k^3 * x^(k^2) / (1 - x^(k^2)). - _Ilya Gutkovskiy_, Jun 05 2024
%p A361794 gsigma := proc(n,z,k)
%p A361794     local a,d ;
%p A361794     a := 0 ;
%p A361794     for d in numtheory[divisors](n) do
%p A361794         if modp(n,d^k) = 0 then
%p A361794             a := a+d^z ;
%p A361794         end if ;
%p A361794     end do:
%p A361794     a ;
%p A361794 end proc:
%p A361794 seq( gsigma(n,3,2),n=1..80) ;
%t A361794 f[p_, e_] := (p^(3*(Floor[e/2] + 1)) - 1)/(p^3 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Mar 24 2023 *)
%o A361794 (PARI) a(n) = sumdiv(n, d, if (issquare(d), sqrtint(d)^3)); \\ _Michel Marcus_, Mar 24 2023
%o A361794 (Python)
%o A361794 from math import prod
%o A361794 from sympy import factorint
%o A361794 def A361794(n): return prod((p**(3*(e>>1)+3)-1)//(p**3-1) for p, e in factorint(n).items()) # _Chai Wah Wu_, Mar 24 2023
%Y A361794 Cf. A010052, A035316, A069290, A351308, A333843, A333844.
%K A361794 nonn,mult,easy
%O A361794 1,4
%A A361794 _R. J. Mathar_, Mar 24 2023