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%I #36 Sep 15 2020 06:35:55
%S 1,2,2,4,2,4,2,5,4,4,2,8,2,4,4,7,2,8,2,8,4,4,2,10,4,4,5,8,2,8,2,8,4,4,
%T 4,16,2,4,4,10,2,8,2,8,8,4,2,14,4,8,4,8,2,10,4,10,4,4,2,16,2,4,8,10,4,
%U 8,2,8,4,8,2,20,2,4,8,8,4,8,2,14,7,4,2,16,4,4,4,10,2,16,4,8,4,4,4,16
%N Number of square divisors of n-th cube: a(n) = A046951(n^3).
%C Apparently the inverse Mobius transform of A056624 (and therefore multiplicative). - _R. J. Mathar_, Feb 07 2011
%H Antti Karttunen, <a href="/A092520/b092520.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = A000005(n) iff n is squarefree.
%F From _Werner Schulte_, Feb 19 2018: (Start)
%F Multiplicative with a(p^e) = floor((3*e+2)/2) = A001651(e+1), p prime and e >= 0.
%F Dirichlet g.f.: Sum_{n>0} a(n)/n^s = (zeta(s))^2 * zeta(2*s) / zeta(3*s). (End)
%F Sum_{k=1..n} a(k) ~ Pi^2 * n/(6*Zeta(3)) * (log(n) - 1 + 2*gamma + 12*Zeta'(2)/Pi^2 - 3*Zeta'(3)/Zeta(3)) + Zeta(1/2)^2 * sqrt(n) / Zeta(3/2), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Feb 08 2019
%e n=12, the divisors of 12^3 = 1728 are 1=1^2, 2, 3, 4=2^2, 6, 8, 9=3^2, 12, 16=4^2, 18, 24, 27, 32, 36=6^2, 48, 54, 64=8^2, 72, 96, 108, 144=12^2, 192, 216, 288, 432, 576=24^2, 864 and 1728: eight of them are squares, therefore a(12) = 8.
%p A092520 := proc(n)
%p A046951(n^3) ;
%p end proc:
%p seq(A092520(n),n=1..50) ; # _R. J. Mathar_, Jul 09 2016
%t a[n_] := Count[Divisors[n^3], d_ /; IntegerQ[Sqrt[d]]]; Array[a, 100] (* _Jean-François Alcover_, Feb 13 2018 *)
%t f[p_, e_] := Floor[(3*e+2)/2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 15 2020 *)
%o (PARI) A046951(n) = factorback(apply(e->e\2+1, factor(n)[, 2])) \\ This function from _Charles R Greathouse IV_, Sep 17 2015
%o A092520(n) = A046951(n^3); \\ _Antti Karttunen_, May 25 2017
%o (PARI) a(n) = sumdiv(n^3, d, issquare(d)); \\ _Michel Marcus_, Apr 08 2018
%Y Cf. A000005, A000290, A000578, A005117, A046951, A048785, A056624.
%K nonn,mult
%O 1,2
%A _Reinhard Zumkeller_, Apr 06 2004
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