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A063775 Number of 4th powers dividing n. 10
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).
Sequence in context: A194333 A203640 A043289 * A053164 A365333 A295658
KEYWORD
mult,easy,nonn
AUTHOR
Henry Bottomley, Aug 16 2001
STATUS
approved

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Last modified May 23 14:50 EDT 2024. Contains 372763 sequences. (Running on oeis4.)