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A298735 Number of odd squares dividing n. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,9
COMMENTS
The smallest integer with exactly m odd square divisors is A357450(m). - Bernard Schott, Oct 03 2022
LINKS
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
Sequence in context: A340102 A031260 A360157 * A055090 A290106 A359433
KEYWORD
nonn,mult,easy
AUTHOR
Ilya Gutkovskiy, Jan 25 2018
EXTENSIONS
Keyword mult added by Rémy Sigrist, Jan 26 2018
STATUS
approved

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Last modified March 28 15:28 EDT 2024. Contains 371254 sequences. (Running on oeis4.)