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

1,8

Comments

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

Links

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

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

Example

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.

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 *)

Prog

(Haskell)
a048105 n = length [d | d <- [1..n], mod n d == 0, gcd d (n `div` d) > 1]
-- Reinhard Zumkeller, Aug 17 2011
(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<<len(f:=factorint(n).values()))+prod(e+1 for e in f) # Chai Wah Wu, Aug 12 2024

Crossrefs

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

Cf. A001620, A229099, A306016.

Sequence in context: A054673 A155103 A295819 * A360012 A363806 A335021

Adjacent sequences: A048102 A048103 A048104 * A048106 A048107 A048108

Keyword

nonn

Author

Labos Elemer

Status

approved