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A300909
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Sum of 4th powers dividing n.
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4
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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
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OFFSET
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1,16
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COMMENTS
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Multiplicative with a(p^e) = (p^(4*(1+floor(e/4)))-1)/(p^4-1). - Robert Israel, Mar 15 2018
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LINKS
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FORMULA
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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.
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EXAMPLE
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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 + ...
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MAPLE
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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:
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MATHEMATICA
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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]]
f[p_, e_] := (p^(4*(1 + Floor[e/4])) - 1)/(p^4 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 01 2020 *)
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PROG
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(PARI) a(n) = sumdiv(n, d, d*ispower(d, 4)); \\ Michel Marcus, Mar 15 2018
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CROSSREFS
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KEYWORD
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nonn,mult
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AUTHOR
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STATUS
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approved
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