A326988 Sum of nonpowers of 2 dividing n.

0, 0, 3, 0, 5, 9, 7, 0, 12, 15, 11, 21, 13, 21, 23, 0, 17, 36, 19, 35, 31, 33, 23, 45, 30, 39, 39, 49, 29, 69, 31, 0, 47, 51, 47, 84, 37, 57, 55, 75, 41, 93, 43, 77, 77, 69, 47, 93, 56, 90, 71, 91, 53, 117, 71, 105, 79, 87, 59, 161, 61, 93, 103, 0, 83, 141, 67, 119, 95, 141, 71, 180, 73, 111, 123, 133, 95, 165, 79, 155

Offset: 1

Author

Omar E. Pol, Aug 18 2019

Keywords

• nonn
• easy

Comments

In other words: a(n) is the sum of the divisors of n that are not powers of 2.

a(n) is also the sum of odd divisors greater than 1 of n, multiplied by the sum of the divisors of n that are powers of 2.

a(n) = 0 if and only if n is a power of 2.

a(n) = n if and only if n is an odd prime.

From Bernard Schott, Sep 17 2019: (Start)

a(n) = 3*n/2 if and only if n is an even semiprime greater than or equal to 6 (A100484).

a(n) = n + sqrt(n) if and only if n is the square of an odd prime (see A001248 without its first term). (End)

Examples

			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]. The sum of them is 3 + 6 + 9 + 18 = 36, so a(18) = 36.
On the other hand, the sum of odd divisors greater than 1 of 18 is 3 + 9 = 12, and the sum of the divisors of 18 that are powers of 2 is 1 + 2 = 3, then we have that 12 * 3 = 36, so a(18) = 36.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Row sums of A326989.

Cf. A000079, A000203, A000593, A038712, A057716, A065091, A326987 (number), A326990.

Programs

Magma

sol:=[];  m:=1;  for n in [1..80] 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

Maple

f:= n -> numtheory:-sigma(n) - 2^(1+padic:-ordp(n,2))+1:
map(f, [$1..100]); # Robert Israel, Apr 29 2020

Mathematica

Table[DivisorSigma[1, n] - Denominator[DivisorSigma[1, 2n]/DivisorSigma[1, n]], {n, 100}] (* Wesley Ivan Hurt, Aug 24 2019 *)

PARI

ispp2(n) = (n==1) || (isprimepower(n, &p) && (p==2));
a(n) = sumdiv(n, d, if (!ispp2(d), d)); \\ Michel Marcus, Aug 26 2019

Python

from sympy import divisor_sigma
def A326988(n): return divisor_sigma(n)-(n^(n-1)) # Chai Wah Wu, Aug 04 2022

Scala

def divisors(n: Int): IndexedSeq[Int] = (1 to n).filter(n % _ == 0)
(1 to 80).map(divisors().filter(n => n != Integer.highestOneBit(n)).sum) // _Alonso del Arte, Apr 29 2020

Formula

a(n) = A000203(n) - A038712(n).

a(n) = (A000593(n) - 1)*A038712(n).

a(n) = A326990(n)*A038712(n).

a(n) = Sum_{d|n, d > 1} d * (1 - [rad(d) = 2]), where rad is the squarefree kernel (A007947) and [] is the Iverson bracket, which gives 1 if the condition is true, 0 if it's false. - Wesley Ivan Hurt, Apr 29 2020