cp's OEIS Frontend

A048106 Number of unitary divisors of n (A034444) - number of non-unitary divisors of n (A048105).

%I A048106 #22 Dec 09 2022 07:07:08
%S A048106 1,2,2,1,2,4,2,0,1,4,2,2,2,4,4,-1,2,2,2,2,4,4,2,0,1,4,0,2,2,8,2,-2,4,
%T A048106 4,4,-1,2,4,4,0,2,8,2,2,2,4,2,-2,1,2,4,2,2,0,4,0,4,4,2,4,2,4,2,-3,4,8,
%U A048106 2,2,4,8,2,-4,2,4,2,2,4,8,2,-2,-1,4,2,4,4,4,4,0,2,4,4,2,4,4,4,-4,2,2,2
%N A048106 Number of unitary divisors of n (A034444) - number of non-unitary divisors of n (A048105).
%H A048106 Antti Karttunen, <a href="/A048106/b048106.txt">Table of n, a(n) for n = 1..10000</a>
%F A048106 a(n) = 2^(1+omega(n)) - d(n) = 2^(1+A001221(n)) - A000005(n).
%F A048106 a(n) = -Sum_{ d divides n } (-1)^mu(d). - _Vladeta Jovovic_, Jan 24 2002
%F A048106 From _Amiram Eldar_, Dec 09 2022: (Start)
%F A048106 a(n) > 0 iff n is in A048107.
%F A048106 a(n) < 0 iff n is in A048111.
%F A048106 a(n) <= 0 iff n is in A048108.
%F A048106 a(n) = 0 iff n is in A048109.
%F A048106 Dirichlet g.f: zeta(s)^2*(2/zeta(2*s) - 1).
%F A048106 Sum_{k=1..n} a(k) ~ (12/Pi^2 - 1)*n*log(n) + ((12/Pi^2-1)*(2*gamma-1) - (24/Pi^2)*zeta'(2)/zeta(2))*n, where gamma is Euler's constant (A001620). (End)
%t A048106 Table[2^(1 + PrimeNu@ n) - DivisorSigma[0, n], {n, 99}] (* _Michael De Vlieger_, Aug 01 2017 *)
%o A048106 (PARI) A048106(n) = (2^(1+omega(n)) - numdiv(n)); \\ _Antti Karttunen_, May 25 2017
%o A048106 (Python)
%o A048106 from sympy import divisor_count, primefactors
%o A048106 def a(n): return 1 if n==1 else 2**(1 + len(primefactors(n))) - divisor_count(n) # _Indranil Ghosh_, May 25 2017
%Y A048106 Cf. A000005, A001221, A008683, A034444, A048105.
%Y A048106 Cf. A048107, A048108, A048109, A048111.
%Y A048106 Cf. A001620, A306016.
%K A048106 sign
%O A048106 1,2
%A A048106 _Labos Elemer_