

A326987


Number of nonpowers of 2 dividing n.


6



0, 0, 1, 0, 1, 2, 1, 0, 2, 2, 1, 3, 1, 2, 3, 0, 1, 4, 1, 3, 3, 2, 1, 4, 2, 2, 3, 3, 1, 6, 1, 0, 3, 2, 3, 6, 1, 2, 3, 4, 1, 6, 1, 3, 5, 2, 1, 5, 2, 4, 3, 3, 1, 6, 3, 4, 3, 2, 1, 9, 1, 2, 5, 0, 3, 6, 1, 3, 3, 6, 1, 8, 1, 2, 5, 3, 3, 6, 1, 5, 4, 2, 1, 9, 3, 2, 3, 4, 1, 10, 3, 3, 3, 2, 3, 6, 1, 4, 5, 6
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OFFSET

1,6


COMMENTS

In other words: a(n) is the number of divisors of n that are not powers of 2.
a(n) is also the number of odd divisors > 1 of n, multiplied by the number of divisors of n that are powers of 2.
a(n) = 0 iff n is a power of 2.
a(n) = 1 iff n is an odd prime.
From Bernard Schott, Sep 12 2019: (Start)
a(n) = 2 iff n is an even semiprime >= 6 or n is a square of prime >= 9 (Aug 26 2019).
a(n) = 3 iff n is an odd squarefree semiprime, or n is an odd prime multiplied by 4, or n is a cube of odd prime (End).


LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..20000


FORMULA

a(n) = A000005(n)  A001511(n).
a(n) = (A001227(n)  1)*A001511(n).
a(n) = A069283(n)*A001511(n).


EXAMPLE

For n = 18 the divisors of 18 are [1, 2, 3, 6, 9, 18]. There are four divisors of 18 that are not powers of 2, they are [3, 6, 9, 18], so a(18) = 4. On the other hand, there are two odd divisors > 1 of 18, they are [3, 9], and there are two divisors of 18 that are powers of 2, they are [1, 2], then we have that 2*2 = 4, so a(18) = 4.


MAPLE

a:= n> numtheory[tau](n)padic[ordp](2*n, 2):
seq(a(n), n=1..100); # Alois P. Heinz, Aug 24 2019


MATHEMATICA

a[n_] := DivisorSigma[0, n]  IntegerExponent[n, 2]  1; Array[a, 100] (* Amiram Eldar, Aug 31 2019 *)


PROG

(MAGMA) sol:=[]; m:=1; for n in [1..100] do v:=Set(Divisors(n)) diff {2^k:k in [0..Floor(Log(2, n))]}; sol[m]:=#v; m:=m+1; end for; sol; // Marius A. Burtea, Aug 24 2019
(PARI) ispp2(n) = (n==1)  (isprimepower(n, &p) && (p==2));
a(n) = sumdiv(n, d, ispp2(d) == 0); \\ Michel Marcus, Aug 26 2019


CROSSREFS

Cf. A000005, A000079, A001227, A001248, A001511, A057716, A065091, A069283, A100484, A326988 (sum), A326989.
Sequence in context: A294508 A035152 A035204 * A190775 A282459 A016154
Adjacent sequences: A326984 A326985 A326986 * A326988 A326989 A326990


KEYWORD

nonn,easy


AUTHOR

Omar E. Pol, Aug 18 2019


STATUS

approved



