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A065704 Number of squares or twice squares dividing n. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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 *)

CROSSREFS

Cf. A000203, A028982, A046951.

Sequence in context: A194550 A339914 A242923 * 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

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Last modified April 10 16:05 EDT 2021. Contains 342845 sequences. (Running on oeis4.)