A048105 - OEIS

%I #39 Aug 13 2024 01:56:13

%S 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,

%T 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,

%U 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

%N Number of non-unitary divisors of n.

%C Number of zeros in row n of table A225817. - _Reinhard Zumkeller_, Jul 30 2013

%H Reinhard Zumkeller, <a href="/A048105/b048105.txt">Table of n, a(n) for n = 1..10000</a>

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

%F From _Reinhard Zumkeller_, Jul 30 2013: (Start)

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

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

%F Dirichlet g.f.: zeta(s)^2 - zeta(s)^2/zeta(2*s). - _Geoffrey Critzer_, Dec 10 2014

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

%F 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

%e Example 1: If n is squarefree (A005117) then a(n)=0 since all divisors are unitary.

%e 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.

%e 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.

%p with(NumberTheory):

%p seq(SumOfDivisors(n, 0) - 2^NumberOfPrimeFactors(n, 'distinct'), n = 1..105);

%p # _Peter Luschny_, Jul 27 2023

%t Table[DivisorSigma[0, n] - 2^PrimeNu[n], {n, 1, 50}] (* _Geoffrey Critzer_, Dec 10 2014 *)

%o (Haskell)

%o a048105 n = length [d | d <- [1..n], mod n d == 0, gcd d (n `div` d) > 1]

%o -- _Reinhard Zumkeller_, Aug 17 2011

%o (PARI) a(n)=my(f=factor(n)[,2]); prod(i=1,#f,f[i]+1)-2^#f \\ _Charles R Greathouse IV_, Sep 18 2015

%o (Python)

%o from math import prod

%o from sympy import factorint

%o def A048105(n): return -(1<<len(f:=factorint(n).values()))+prod(e+1 for e in f) # _Chai Wah Wu_, Aug 12 2024

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

%Y Cf. A001620, A229099, A306016.

%K nonn

%O 1,8

%A _Labos Elemer_