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A063775 Number of 4th powers dividing n. 7
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 (list; graph; refs; listen; history; text; internal format)
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 A295658 A307427

Adjacent sequences:  A063772 A063773 A063774 * A063776 A063777 A063778

KEYWORD

mult,easy,nonn

AUTHOR

Henry Bottomley, Aug 16 2001

STATUS

approved

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Last modified September 28 10:48 EDT 2021. Contains 347714 sequences. (Running on oeis4.)