A048105 Number of non-unitary divisors of n.

0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 2, 0, 0, 0, 3, 0, 2, 0, 2, 0, 0, 0, 4, 1, 0, 2, 2, 0, 0, 0, 4, 0, 0, 0, 5, 0, 0, 0, 4, 0, 0, 0, 2, 2, 0, 0, 6, 1, 2, 0, 2, 0, 4, 0, 4, 0, 0, 0, 4, 0, 0, 2, 5, 0, 0, 0, 2, 0, 0, 0, 8, 0, 0, 2, 2, 0, 0, 0, 6, 3, 0, 0, 4, 0, 0, 0, 4, 0, 4, 0, 2, 0, 0, 0, 8, 0, 2, 2, 5, 0, 0, 0, 4, 0

Offset: 1

Author

Labos Elemer

Keywords

• nonn

Comments

Number of zeros in row n of table A225817. - Reinhard Zumkeller, Jul 30 2013

Examples

			Example 1: If n is squarefree (A005117) then a(n)=0 since all divisors are unitary.
Example 2: n=12, d(n)=6, ud(n)=4, nud(12)=d(12)-ud(12)=2; from {1,2,3,4,6,12} {1,3,4,12} are unitary while {2,6} are not unitary divisors.
Example 3: n=p^k, a true prime power, d(n)=k+1, u(d)=2^r(x)=2, so nud(n)=d(p^k)-2=k+1 i.e., it can be arbitrarily large.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A001221, A007875, A056170, A225817, A034444.

Cf. A001620, A229099, A306016.

Programs

Haskell

a048105 n = length [d | d <- [1..n], mod n d == 0, gcd d (n `div` d) > 1]
-- Reinhard Zumkeller, Aug 17 2011

Maple

with(NumberTheory):
seq(SumOfDivisors(n, 0) - 2^NumberOfPrimeFactors(n, 'distinct'), n = 1..105);
# Peter Luschny, Jul 27 2023

Mathematica

Table[DivisorSigma[0, n] - 2^PrimeNu[n], {n, 1, 50}] (* Geoffrey Critzer, Dec 10 2014 *)

PARI

a(n)=my(f=factor(n)[,2]); prod(i=1,#f,f[i]+1)-2^#f \\ Charles R Greathouse IV, Sep 18 2015

Python

from math import prod
from sympy import factorint
def A048105(n): return -(1<Chai Wah Wu, Aug 12 2024

Formula

a(n) = Sigma(0, n) - 2^r(n), where r() = A001221, the number of distinct primes dividing n.

From Reinhard Zumkeller, Jul 30 2013: (Start)

a(n) = A000005(n) - A034444(n).

For n > 1: a(n) = A000005(n) - 2 * A007875(n). (End)

Dirichlet g.f.: zeta(s)^2 - zeta(s)^2/zeta(2*s). - Geoffrey Critzer, Dec 10 2014

G.f.: Sum_{k>=1} (1 - mu(k)^2)*x^k/(1 - x^k). - Ilya Gutkovskiy, Apr 21 2017

Sum_{k=1..n} a(k) ~ (1-6/Pi^2)*n*log(n) + ((1-6/Pi^2)*(2*gamma-1)+(72*zeta'(2)/Pi^4))*n , where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 27 2022