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)
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
KEYWORD
nonn
AUTHOR
STATUS
approved