A298735 Number of odd squares dividing n.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1

Offset

1,9

Comments

The smallest integer with exactly m odd square divisors is A357450(m). - Bernard Schott, Oct 03 2022

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

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

Multiplicative with a(2^e) = 1 and a(p^e) = floor(e/2) + 1 for p > 2. - Amiram Eldar, Sep 11 2020

a(n) = A046951(4*n) - A046951(n) = A046951(A000265(n)). - Velin Yanev, Antti Karttunen, Dec 06 2021

Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi^2/8 (A111003). - Amiram Eldar, Sep 25 2022

Example

a(81) = 3 because 81 has 5 divisors {1, 3, 9, 27, 81} among which 3 are odd squares {1, 9, 81}.

Mathematica

nmax = 105; Rest[CoefficientList[Series[Sum[x^(2 k - 1)^2/(1 - x^(2 k - 1)^2), {k, 1, nmax}], {x, 0, nmax}], x]]
a[n_] := Length[Select[Divisors[n], IntegerQ[Sqrt[#]] && OddQ[#] &]]; Table[a[n], {n, 1, 105}]
f[2, e_] := 1; f[p_, e_] := Floor[e/2] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 11 2020 *)

Prog

(PARI) a(n)=factorback(apply(e->e\2+1, factor(n/2^valuation(n, 2))[, 2])) \\ Rémy Sigrist, Jan 26 2018

Crossrefs

Cf. A000265, A001227, A016754, A046951, A056170, A071325, A111003, A122132 (positions of ones), A357450.

Sequence in context: A340102 A031260 A360157 * A055090 A290106 A359433

Adjacent sequences: A298732 A298733 A298734 * A298736 A298737 A298738

Keyword

nonn,mult,easy

Author

Ilya Gutkovskiy, Jan 25 2018

Extensions

Keyword mult added by Rémy Sigrist, Jan 26 2018

Status

approved