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A304906
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L.g.f.: log(Product_{k>=1} (1 + x^(k^3))) = Sum_{n>=1} a(n)*x^n/n.
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2
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1, -1, 1, -1, 1, -1, 1, 7, 1, -1, 1, -1, 1, -1, 1, -9, 1, -1, 1, -1, 1, -1, 1, 7, 1, -1, 28, -1, 1, -1, 1, -9, 1, -1, 1, -1, 1, -1, 1, 7, 1, -1, 1, -1, 1, -1, 1, -9, 1, -1, 1, -1, 1, -28, 1, 7, 1, -1, 1, -1, 1, -1, 1, 55, 1, -1, 1, -1, 1, -1, 1, 7, 1, -1, 1, -1, 1, -1, 1, -9, 28, -1
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OFFSET
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1,8
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LINKS
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FORMULA
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G.f.: Sum_{k>=1} k^3*x^(k^3)/(1 + x^(k^3)).
a(n) = Sum_{d|n} (-1)^(n/d+1)*A010057(d)*d.
a(n) = 1 if n is an odd cubefree.
a(n) = -1 if n is an even cubefree.
Multiplicative with a(p^e) = (-1)^(p+1) * (p^(3*floor((e+3)/3))-1)/(p^3-1) + if(p = 2 and e == 0 (mod 3), 2^(e + 1) otherwise 0). - Amiram Eldar, Oct 03 2023
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EXAMPLE
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L.g.f.: L(x) = x - x^2/2 + x^3/3 - x^4/4 + x^5/5 - x^6/6 + x^7/7 + 7*x^8/8 + x^9/9 - x^10/10 + x^11/11 - x^12/12 + ...
exp(L(x)) = 1 + x + x^8 + x^9 + x^27 + x^28 + x^35 + x^36 + x^64 + x^65 + x^72 + x^73 + ... + A279329(n)*x^n + ...
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MATHEMATICA
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nmax = 82; Rest[CoefficientList[Series[Log[Product[1 + x^k^3, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
nmax = 82; Rest[CoefficientList[Series[Sum[k^3 x^k^3/(1 + x^k^3), {k, 1, Floor[nmax^(1/3) + 1]}], {x, 0, nmax}], x]]
Table[DivisorSum[n, (-1)^(n/# + 1) # &, IntegerQ[#^(1/3)] &], {n, 82}]
f[p_, e_] := (-1)^(p+1) * (p^(3*Floor[(e+3)/3])-1)/(p^3-1) + If[p == 2 && Divisible[e, 3], 2^(e + 1), 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 03 2023 *)
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PROG
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(PARI) seq(n)={Vec(deriv(log(prod(k=1, sqrtnint(n, 3), 1 + x^(k^3) + O(x*x^n)))))} \\ Andrew Howroyd, Jul 20 2018
(PARI) a(n)={sumdiv(n, d, if(n%d^3, 0, (-1)^(n/d^3 + 1) * d^3))} \\ Andrew Howroyd, Jul 20 2018
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CROSSREFS
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KEYWORD
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sign,easy,mult
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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