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A063775 Number of 4th powers dividing n.

%I A063775 #39 Sep 15 2020 06:36:08
%S A063775 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,
%T A063775 1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,
%U A063775 1,1,1,1,1,1,1,1,1,1,1,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1
%N A063775 Number of 4th powers dividing n.
%H A063775 Amiram Eldar, <a href="/A063775/b063775.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..2000 from Harry J. Smith)
%F A063775 a(n) = A000005(A053164(n)) = A046951(A000188(n)).
%F A063775 Multiplicative with a(p^e) = 1 + floor(e/4).
%F A063775 Dirichlet g.f.: zeta^2(4s)*Product_{primes p} (1 + p^(-s) + p^(-2s) + p^(-3s)). - _R. J. Mathar_, Jan 11 2012
%F A063775 G.f.: Sum_{k>=1} x^(k^4)/(1 - x^(k^4)). - _Ilya Gutkovskiy_, Mar 21 2017
%F A063775 Dirichlet g.f.: zeta(s) * zeta(4s). - _Álvar Ibeas_, Dec 29 2018
%F A063775 Sum_{k=1..n} a(k) ~ Pi^4 * n / 90 + Zeta(1/4) * n^(1/4). - _Vaclav Kotesovec_, Feb 03 2019
%e A063775 a(79) = 1 since 79 is divisible by 1 = 1^4.
%e A063775 a(80) = 2 since 80 is divisible by 1 and 16 = 2^4.
%e A063775 a(81) = 2 since 81 is divisible by 1 and 81 = 3^4.
%p A063775 seq(coeff(series(add(x^(k^4)/(1-x^(k^4)),k=1..n),x,n+1), x, n), n = 1 .. 120); # _Muniru A Asiru_, Dec 29 2018
%t A063775 nn = 100;f[list_, i_] := list[[i]];
%t A063775 Table[DirichletConvolve[f[Boole[Map[IntegerQ[#] &, Map[#^(1/4) &,Range[nn]]]], n],f[Table[1, {nn}], n], n, m], {m, 1, nn}] (* _Geoffrey Critzer_, Feb 07 2015 *)
%t A063775 f[p_, e_] := 1 + Floor[e/4]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Sep 15 2020 *)
%o A063775 (PARI) { for (n=1, 2000, k=2; a=1; while ((p=k^4) <= n, if (n%p == 0, a++); k++); write("b063775.txt", n, " ", a) ) } \\ _Harry J. Smith_, Aug 30 2009
%Y A063775 Cf. A046951 (number of squares), A061704 (number of cubes).
%Y A063775 Cf. A000005, A053164, A046951, A000188.
%K A063775 mult,easy,nonn
%O A063775 1,16
%A A063775 _Henry Bottomley_, Aug 16 2001