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A327251
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Expansion of Sum_{k>=1} psi(k) * x^k / (1 - x^k)^2, where psi = A001615.
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6
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1, 5, 7, 16, 11, 35, 15, 44, 33, 55, 23, 112, 27, 75, 77, 112, 35, 165, 39, 176, 105, 115, 47, 308, 85, 135, 135, 240, 59, 385, 63, 272, 161, 175, 165, 528, 75, 195, 189, 484, 83, 525, 87, 368, 363, 235, 95, 784, 161, 425, 245, 432, 107, 675, 253, 660, 273
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OFFSET
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1,2
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COMMENTS
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Inverse Moebius transform of A322577.
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LINKS
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FORMULA
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a(n) = Sum_{d|n} psi(n/d) * d.
a(p) = 2*p + 1, where p is prime.
Multiplicative with a(p^e) = p^(e-1)*((p+1)*e + p). - Antti Karttunen, Aug 24 2021
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MATHEMATICA
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nmax = 57; CoefficientList[Series[Sum[DirichletConvolve[j, MoebiusMu[j]^2, j, k] x^k/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
f[p_, e_] := p^(e - 1)*((p + 1)*e + p); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 24 2021 *)
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PROG
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(PARI) mypsi(n) = n * sumdivmult(n, d, issquarefree(d)/d); \\ A001615
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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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