A063775 Number of 4th powers dividing n.

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, 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, 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

Offset

1,16

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2000 from Harry J. Smith)

Formula

a(n) = A000005(A053164(n)) = A046951(A000188(n)).

Multiplicative with a(p^e) = 1 + floor(e/4).

Dirichlet g.f.: zeta^2(4s)*Product_{primes p} (1 + p^(-s) + p^(-2s) + p^(-3s)). - R. J. Mathar, Jan 11 2012

G.f.: Sum_{k>=1} x^(k^4)/(1 - x^(k^4)). - Ilya Gutkovskiy, Mar 21 2017

Dirichlet g.f.: zeta(s) * zeta(4s). - Álvar Ibeas, Dec 29 2018

Sum_{k=1..n} a(k) ~ Pi^4 * n / 90 + Zeta(1/4) * n^(1/4). - Vaclav Kotesovec, Feb 03 2019

Example

a(79) = 1 since 79 is divisible by 1 = 1^4.
a(80) = 2 since 80 is divisible by 1 and 16 = 2^4.
a(81) = 2 since 81 is divisible by 1 and 81 = 3^4.

Maple

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

Mathematica

nn = 100; f[list_, i_] := list[[i]];
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 *)
f[p_, e_] := 1 + Floor[e/4]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 15 2020 *)

Prog

(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

Crossrefs

Cf. A046951 (number of squares), A061704 (number of cubes).

Cf. A000005, A053164, A046951, A000188.

Sequence in context: A194333 A203640 A043289 * A053164 A365333 A295658

Adjacent sequences: A063772 A063773 A063774 * A063776 A063777 A063778

Keyword

mult,easy,nonn

Author

Henry Bottomley, Aug 16 2001

Status

approved