A326989 -id:A326989 - cp's OEIS frontend

A326987 Number of nonpowers of 2 dividing n.

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

Offset: 1

Omar E. Pol, Aug 18 2019

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).

			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.

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

Cf. A000005, A000079, A001227, A001248, A001511, A001620, A057716, A065091, A069283, A100484, A326988 (sum), A326989.

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

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

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

ispp2(n) = (n==1) || (isprimepower(n, &p) && (p==2));
a(n) = sumdiv(n, d, ispp2(d) == 0); \\ Michel Marcus, Aug 26 2019

from sympy import divisor_count
def A326987(n): return divisor_count(n)-(n&-n).bit_length() # Chai Wah Wu, Jul 13 2022

a(n) = A000005(n) - A001511(n).
a(n) = (A001227(n) - 1)*A001511(n).
a(n) = A069283(n)*A001511(n).
Sum_{k=1..n} a(k) ~ n * (log(n) + 2*gamma - 3), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 18 2024

A326988 Sum of nonpowers of 2 dividing n.

Original entry on oeis.org

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

Omar E. Pol, Aug 18 2019

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)

			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.

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

Row sums of A326989.

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

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

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

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

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

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

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

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

A326990 Sum of odd divisors of n that are greater than 1.

Original entry on oeis.org

0, 0, 3, 0, 5, 3, 7, 0, 12, 5, 11, 3, 13, 7, 23, 0, 17, 12, 19, 5, 31, 11, 23, 3, 30, 13, 39, 7, 29, 23, 31, 0, 47, 17, 47, 12, 37, 19, 55, 5, 41, 31, 43, 11, 77, 23, 47, 3, 56, 30, 71, 13, 53, 39, 71, 7, 79, 29, 59, 23, 61, 31, 103, 0, 83, 47, 67, 17, 95, 47, 71, 12, 73, 37, 123, 19, 95, 55, 79, 5

Offset: 1

Omar E. Pol, Aug 24 2019

nonn

Also sum of nonpowers of 2 dividing n, divided the sum of powers of 2 dividing n.
a(n) = 0 iff n is a power of 2.
a(n) = n iff n is an odd prime.
First differs from A284233 at a(15).

			For n = 18 the divisors of 18 are [1, 2, 3, 6, 9, 18]. The sum of odd divisors of 18 that are greater than 1 is 3 + 9 = 12, so a(18) = 12. On the other hand, there are four divisors of 18 that are not powers of 2, they are [3, 6, 9, 18], and the sum of them is 3 + 6 + 9, 18 = 36. Also there are two divisors of 18 that are powers of 2, they are [1, 2], and the sum of them is 1 + 2 = 3. Then we have that 36/3 = 12, so a(18) = 12.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000079, A000203, A000593, A038712, A057716, A065091, A069283 (Number), A284233, A326988, A326989.

sol:=[]; m:=1; for n in [1..80] do v:=[d:d in Divisors(n)|d gt 1 and IsOdd(d)]; if #v ne 0 then sol[m]:=&+v; m:=m+1; else sol[m]:=0; m:=m+1; end if; end for; sol; // Marius A. Burtea, Aug 24 2019

Table[Total[Select[Rest[Divisors[n]],OddQ]],{n,80}] (* Harvey P. Dale, Jan 11 2025 *)

a(n) = A000593(n) - 1.
a(n) = (A000203(n) - A038712(n))/A038712(n).
a(n) = A326988(n)/A038712(n).

A327328 a(n) is the smallest positive integer divisible by exactly n nonpowers of 2.

Original entry on oeis.org

1, 3, 6, 12, 18, 45, 30, 105, 72, 60, 90, 315, 120, 3645, 210, 180, 450, 1575, 480, 2835, 360, 420, 630, 3465, 900, 720, 7290, 1620, 840, 14175, 1440, 10395, 1800, 1260, 3150, 1680, 3240, 1937102445, 5670, 14580, 3600, 127575, 3360, 2066715, 2520, 3780, 6930

Offset: 0

Omar E. Pol, Sep 20 2019

nonn

Terms are of the form A233819(m) * 2^k for some m > 0, k >= 0. - David A. Corneth, Nov 25 2019

David A. Corneth, Table of n, a(n) for n = 0..1000

Cf. A057716, A233819, A326987, A326989.

ispp(x) = my(p); (x == 1) || (isprimepower(x, &p) && (p==2));
nbdiv(k) = #select(x->(!ispp(x)), divisors(k));
a(n) = my(k=1); while (nbdiv(k) != n, k++); k; \\ Michel Marcus, Nov 25 2019

a(n) = for(i = 1, oo, if(numdiv(i) - valuation(i, 2) - 1 == n, return(i))) \\ David A. Corneth, Nov 25 2019

a(13)-a(36), a(38)-a(46) from Jon E. Schoenfield, Nov 22 2019

a(37) from Jon E. Schoenfield and David A. Corneth, Nov 25 2019

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A326987 Number of nonpowers of 2 dividing n.

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A326988 Sum of nonpowers of 2 dividing n.

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A326990 Sum of odd divisors of n that are greater than 1.

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A327328 a(n) is the smallest positive integer divisible by exactly n nonpowers of 2.

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