A344299 Expansion of Sum_{k>=1} (-1)^(k+1) * x^(k^2) / (1 - x^(k^2)).

1, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 1, -1, 1, 2, 1, 0, 1, 1, 1, 0, 2, 1, 2, 0, 1, 1, 1, -1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, -1, 2, 2, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 1, 1, 2, -2, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, -1, 3, 1, 1, 0, 1, 1, 1, 0, 1, 2

Offset

1,9

Comments

Number of odd squares dividing n minus number of even squares dividing n.

Inverse Moebius transform of A258998.

Links

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

Formula

G.f.: Sum_{k>=1} (1 - theta_4(x^k)) / 2.

From Amiram Eldar, Nov 15 2022: (Start)

Multiplicative with a(2^e) = 1 - floor(e/2), and a(p^e) = 1 + floor(e/2) for p > 2.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi^2/12 (A072691). (End)

Mathematica

nmax = 90; CoefficientList[Series[Sum[(-1)^(k + 1) x^(k^2)/(1 - x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (-1)^(# + 1) &, IntegerQ[#^(1/2)] &], {n, 1, 90}]
f[p_, e_] := 1 - (-1)^p*Floor[e/2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 15 2022 *)

Prog

(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 - (-1)^f[i, 1] * floor(f[i, 2]/2)); } \\ Amiram Eldar, Nov 15 2022

Crossrefs

Cf. A046951, A048272, A072691, A258998, A317529, A344300.

Sequence in context: A367623 A336676 A135341 * A033665 A104234 A355905

Adjacent sequences: A344296 A344297 A344298 * A344300 A344301 A344302

Keyword

sign,mult

Author

Ilya Gutkovskiy, May 14 2021

Status

approved