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A326987
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Number of nonpowers of 2 dividing n.
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6
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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
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1,6
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COMMENTS
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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.
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).
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LINKS
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FORMULA
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Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 3), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 18 2024
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EXAMPLE
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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.
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MAPLE
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a:= n-> numtheory[tau](n)-padic[ordp](2*n, 2):
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MATHEMATICA
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a[n_] := DivisorSigma[0, n] - IntegerExponent[n, 2] - 1; Array[a, 100] (* Amiram Eldar, Aug 31 2019 *)
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PROG
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(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
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CROSSREFS
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Cf. A000005, A000079, A001227, A001248, A001511, A001620, A057716, A065091, A069283, A100484, A326988 (sum), A326989.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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