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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). 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

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