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

1,3

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)

Links

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

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

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]. 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.

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 *)

Prog

(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
(PARI) ispp2(n) = (n==1) || (isprimepower(n, &p) && (p==2));
a(n) = sumdiv(n, d, if (!ispp2(d), d)); \\ Michel Marcus, Aug 26 2019
(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
(Python)
from sympy import divisor_sigma
def A326988(n): return divisor_sigma(n)-(n^(n-1)) # Chai Wah Wu, Aug 04 2022

Crossrefs

Row sums of A326989.

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

Sequence in context: A076296 A260934 A143280 * A088521 A022837 A328631

Adjacent sequences: A326985 A326986 A326987 * A326989 A326990 A326991

Keyword

nonn,easy

Author

Omar E. Pol, Aug 18 2019

Status

approved