A327122 Expansion of Sum_{k>=1} sigma(k) * x^k / (1 + x^(2*k)), where sigma = A000203.

1, 3, 3, 7, 7, 9, 7, 15, 10, 21, 11, 21, 15, 21, 21, 31, 19, 30, 19, 49, 21, 33, 23, 45, 38, 45, 30, 49, 31, 63, 31, 63, 33, 57, 49, 70, 39, 57, 45, 105, 43, 63, 43, 77, 70, 69, 47, 93, 50, 114, 57, 105, 55, 90, 77, 105, 57, 93, 59, 147, 63, 93, 70, 127, 105

Offset

1,2

Comments

Inverse Moebius transform of A050469.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = Sum_{d|n} A050469(d).

From Amiram Eldar, Aug 28 2023: (Start)

Multiplicative with a(2^e) = 2^(e+1)-1, and if p is an odd prime a(p^e) = (p^(e+2)-(e+2)*p+e+1)/(p-1)^2 if p == 1 (mod 4) and (2*p^(e+2) + ((-1)^e-1)*p - ((-1)^e+1))/(2*(p^2-1)) otherwise.

Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/12 * (A175647/A243381) = 0.753351504961... . (End)

Mathematica

nmax = 65; CoefficientList[Series[Sum[DivisorSigma[1, k] x^k/(1 + x^(2 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
A050469[n_] := DivisorSum[n, # &, MemberQ[{1}, Mod[n/#, 4]] &] - DivisorSum[n, # &, MemberQ[{3}, Mod[n/#, 4]] &]; a[n_] := DivisorSum[n, A050469[#] &]; Table[a[n], {n, 1, 65}]
f[p_, e_] := If[Mod[p, 4] == 1, (p^(e+2)-(e+2)*p+e+1)/(p-1)^2, (2*p^(e+2) + ((-1)^e-1)*p - ((-1)^e+1))/(2*(p^2-1))]; f[2, e_] := 2^(e+1)-1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 70] (* Amiram Eldar, Aug 28 2023 *)

Prog

(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; if(p == 2, 2^(e+1)-1, if(p%4 == 1, (p^(e+2)-(e+2)*p+e+1)/(p-1)^2, (2*p^(e+2) + ((-1)^e-1)*p - ((-1)^e+1))/(2*(p^2-1))))); } \\ Amiram Eldar, Aug 28 2023

Crossrefs

Cf. A000203, A050469, A101455, A167181 (fixed points), A262209, A288417, A327123.

Cf. A072691, A175647, A243381.

Sequence in context: A225851 A107470 A228325 * A071042 A128053 A063212

Adjacent sequences: A327119 A327120 A327121 * A327123 A327124 A327125

Keyword

nonn,easy,mult

Author

Ilya Gutkovskiy, Sep 14 2019

Status

approved