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
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
(Python)
from sympy import divisor_count
def A326987(n): return divisor_count(n)-(n&-n).bit_length() # Chai Wah Wu, Jul 13 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Omar E. Pol, Aug 18 2019
STATUS
approved