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%I #38 Mar 21 2023 15:22:59
%S 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,
%T 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,
%U 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
%N Number of odd squares dividing n.
%C The smallest integer with exactly m odd square divisors is A357450(m). - _Bernard Schott_, Oct 03 2022
%H Antti Karttunen, <a href="/A298735/b298735.txt">Table of n, a(n) for n = 1..65537</a>
%F G.f.: Sum_{k>=1} x^((2*k-1)^2)/(1 - x^((2*k-1)^2)).
%F Multiplicative with a(2^e) = 1 and a(p^e) = floor(e/2) + 1 for p > 2. - _Amiram Eldar_, Sep 11 2020
%F a(n) = A046951(4*n) - A046951(n) = A046951(A000265(n)). - _Velin Yanev_, _Antti Karttunen_, Dec 06 2021
%F Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi^2/8 (A111003). - _Amiram Eldar_, Sep 25 2022
%e a(81) = 3 because 81 has 5 divisors {1, 3, 9, 27, 81} among which 3 are odd squares {1, 9, 81}.
%t nmax = 105; Rest[CoefficientList[Series[Sum[x^(2 k - 1)^2/(1 - x^(2 k - 1)^2), {k, 1, nmax}], {x, 0, nmax}], x]]
%t a[n_] := Length[Select[Divisors[n], IntegerQ[Sqrt[#]] && OddQ[#] &]]; Table[a[n], {n, 1, 105}]
%t 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 *)
%o (PARI) a(n)=factorback(apply(e->e\2+1, factor(n/2^valuation(n,2))[, 2])) \\ _Rémy Sigrist_, Jan 26 2018
%Y Cf. A000265, A001227, A016754, A046951, A056170, A071325, A111003, A122132 (positions of ones), A357450.
%K nonn,mult,easy
%O 1,9
%A _Ilya Gutkovskiy_, Jan 25 2018
%E Keyword mult added by _Rémy Sigrist_, Jan 26 2018
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