A065704 Number of squares or twice squares dividing n.

1, 2, 1, 3, 1, 2, 1, 4, 2, 2, 1, 3, 1, 2, 1, 5, 1, 4, 1, 3, 1, 2, 1, 4, 2, 2, 2, 3, 1, 2, 1, 6, 1, 2, 1, 6, 1, 2, 1, 4, 1, 2, 1, 3, 2, 2, 1, 5, 2, 4, 1, 3, 1, 4, 1, 4, 1, 2, 1, 3, 1, 2, 2, 7, 1, 2, 1, 3, 1, 2, 1, 8, 1, 2, 2, 3, 1, 2, 1, 5, 3, 2, 1, 3, 1, 2, 1, 4, 1, 4, 1, 3, 1, 2, 1, 6, 1, 4, 2, 6, 1, 2, 1, 4, 1

Offset

1,2

Links

Carl R. White, Table of n, a(n) for n = 1..10000

Formula

a(n) = (1/2)*Sum_{ d divides n } (1-(-1)^sigma(d)).

Multiplicative with a(2^e) = e+1 and a(p^e) = floor(e/2)+1 for an odd prime p.

a(n) = A005361(2*n) for n>0 (conjectured). - Werner Schulte, Jan 15 2018 [This is true only for numbers whose odd part (A000265) is cubefree (A004709). Therefore, the least counterexample is n=3^3=27: a(27) = 2 while A005361(2*27) = 3. - Amiram Eldar, Sep 25 2022]

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi^2/4 (A091476). - Amiram Eldar, Sep 25 2022

Example

divisors(36) = {1, 2, 3, 4, 6, 9, 12, 18, 36}, thus a(36) = #{1, 2, 4, 9, 18, 36}=6. a(36) = 1/2*(tau(36)-((-1)^sigma(1)+(-1)^sigma(2)+(-1)^sigma(3)+(-1)^sigma(4)+(-1)^sigma(6)+(-1)^sigma(9)+(-1)^sigma(12)+(-1)^sigma(18)+(-1)^sigma(36))) = 1/2*(9-(-3)) = 6. a(36) = a(2^2*3^2) = a(2^2)*a(3^2) = (2+1)*(1+1) = 6.

Mathematica

f[n_] := Total[1 - (-1)^DivisorSigma[1, Divisors@n]]/2; Array[f, 105] (* Robert G. Wilson v, Jan 02 2013 *)
f[p_, e_] := If[p == 2, e+1, Floor[e/2] + 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 25 2022 *)

Prog

(PARI) a(n) = {my(e = valuation(n, 2), o = n>>e, f = factor(o)); (e+1)*prod(i=1 , #f~, floor(f[i, 2]/2)+1)}; \\ Amiram Eldar, Sep 25 2022

Crossrefs

Cf. A000203, A028982, A046951, A091476.

Cf. A000265, A004709.

Sequence in context: A339914 A242923 A375040 * A325565 A286552 A324826

Adjacent sequences: A065701 A065702 A065703 * A065705 A065706 A065707

Keyword

mult,nonn

Author

Vladeta Jovovic, Dec 04 2001

Extensions

More terms from David Wasserman, Sep 09 2002

Status

approved