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A300909 Sum of 4th powers dividing n. 1
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 82, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,16

COMMENTS

Multiplicative with a(p^e) = (p^(4*(1+floor(e/4)))-1)/(p^4-1). - Robert Israel, Mar 15 2018

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

FORMULA

G.f.: Sum_{k>=1} k^4*x^(k^4)/(1 - x^(k^4)).

L.g.f.: -log(Product_{k>=1} (1 - x^(k^4))) = Sum_{n>=1} a(n)*x^n/n.

D.g.f.: zeta(s)*zeta(4s-4). - Robert Israel, Mar 15 2018

EXAMPLE

a(16) = 17 because 16 has 5 divisors {1, 2, 4, 8, 16} among which 2 divisors {1, 16} are 4th powers and 1 + 16 = 17.

L.g.f.: L(x) = x + x^2/2 + x^3/3 + x^4/4 + x^5/5 + x^6/6 + x^7/7 + x^8/8 + x^9/9 + x^10/10 + x^11/11 + x^12/12 + x^13/13 + x^14/14 + x^15/15 + 17*x^16/16 + x^17/17 + ...

exp(L(x)) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15 + 2*x^16 + 2*x^17 + ... + A046042(n)*x^n + ...

MAPLE

N:= 1000: # for a(1)..a(N)

V:= Vector(N, 1):

for m from 2 to floor(N^(1/4)) do

  R:= [seq(i, i=m^4 .. N, m^4)];

  V[R]:= map(`+`, V[R], m^4)

od:

convert(V, list); # Robert Israel, Mar 15 2018

MATHEMATICA

Table[DivisorSum[n, # &, IntegerQ[#^(1/4)] &], {n, 112}]

nmax = 112; Rest[CoefficientList[Series[Sum[k^4 x^k^4/(1 - x^k^4), {k, 1, 10}], {x, 0, nmax}], x]]

nmax = 112; Rest[CoefficientList[Series[-Log[Product[(1 - x^k^4), {k, 1, 10}]], {x, 0, nmax}], x] Range[0, nmax]]

PROG

(PARI) a(n) = sumdiv(n, d, d*ispower(d, 4)); \\ Michel Marcus, Mar 15 2018

CROSSREFS

Cf. A000583, A001159, A035316, A046042, A046100 (positions of ones), A063775, A113061.

Sequence in context: A144692 A241027 A176728 * A088469 A089170 A040292

Adjacent sequences:  A300906 A300907 A300908 * A300910 A300911 A300912

KEYWORD

nonn,mult

AUTHOR

Ilya Gutkovskiy, Mar 15 2018

STATUS

approved

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Last modified February 19 10:22 EST 2019. Contains 320310 sequences. (Running on oeis4.)