cp's OEIS Frontend

A048109 Numbers having equally many squarefree and nonsquarefree divisors; number of unitary divisors of n (A034444) = number of non-unitary divisors of n (A048105).

%I A048109 #49 Apr 04 2025 06:29:35
%S A048109 8,24,27,40,54,56,88,104,120,125,135,136,152,168,184,189,232,248,250,
%T A048109 264,270,280,296,297,312,328,343,344,351,375,376,378,408,424,440,456,
%U A048109 459,472,488,513,520,536,552,568,584,594,616,621,632,664,680,686,696
%N A048109 Numbers having equally many squarefree and nonsquarefree divisors; number of unitary divisors of n (A034444) = number of non-unitary divisors of n (A048105).
%C A048109 For these terms the number of divisors should be a special power of two because ud(n) = 2^r and nud(n) = ud(n). In particular the exponent of 2 is 1+A001221(n), the number of distinct prime factors + 1. Thus this is a subsequence of A036537 where A000005(A036537(n)) = 2^s; here s = 1+A001221(n).
%C A048109 Let us introduce a function D(n) = sigma_0(n)/2^(alpha(1)+...+alpha(r)), sigma_0(n) number of divisors of n (A000005), prime factorization of n = p(1)^alpha(1) * ... * p(r)^alpha(r), alpha(1)+...+alpha(r) is sequence (A001222). This function splits the set of positive integers into subsets, according to the value of D(n). Squarefree numbers (A005117) has D(n)=1, other numbers are "deviated" from the squarefree ideal and have 0 < D(n) < 1. So for D(n)=1/2 we have A048109, D(n)=3/4 we have A060687. - _Ctibor O. Zizka_, Sep 21 2008
%C A048109 Integers n such that there are exactly 3 Abelian groups of order n. That is, n such that A000688(n)=3. In other words, in the prime factorization of n there is exactly one prime with exponent of 3 and the others have exponent of 1. - _Geoffrey Critzer_, Jun 09 2015
%C A048109 The asymptotic density of this sequence is (6/Pi^2) * Sum_{k>=1} 1/(prime(k)^2*(prime(k)+1)) = (1/zeta(2)) * Sum_{k>=3} (-1)^(k+1) * P(k) = 0.0741777413672596019212880156082745910562809066233004356300970463709875..., where P is the prime zeta function. - _Amiram Eldar_, Jul 11 2020
%H A048109 Robert Israel, <a href="/A048109/b048109.txt">Table of n, a(n) for n = 1..10000</a>
%F A048109 Numbers k such that d(k) = 2^(omega(k)+1) or A000005(k) = 2^(A001221(k) + 1) = 2 * A034444(k).
%e A048109 n = 88 = 2*2*2*11 has 8 divisors, of which 4 are unitary divisors (because of 2 distinct prime factors) and 4 are nonunitary divisors: U={1,88,11,8} and NU = {2,44,4,22}.
%p A048109 filter:= proc(n) local F;
%p A048109   F:= ifactors(n)[2];
%p A048109   mul(t[2]+1,t=F) = 2^(1+nops(F))
%p A048109 end proc;
%p A048109 select(filter, [$1..1000]); # _Robert Israel_, Jun 09 2015
%t A048109 Position[Table[FiniteAbelianGroupCount[n], {n, 1, 1000}],3] // Flatten (* _Geoffrey Critzer_, Jun 09 2015 *)
%o A048109 (PARI) is(n)=select(e->e>1, factor(n)[,2])==[3]~ \\ _Charles R Greathouse IV_, Jun 10 2015
%o A048109 (PARI) isok(n) = sumdiv(n, d, issquarefree(d)) == sumdiv(n, d, !issquarefree(d)); \\ _Michel Marcus_, Jun 24 2015
%o A048109 (Python)
%o A048109 from math import isqrt
%o A048109 from sympy import mobius, primerange
%o A048109 def A048109(n):
%o A048109     def bisection(f,kmin=0,kmax=1):
%o A048109         while f(kmax) > kmax: kmax <<= 1
%o A048109         kmin = kmax >> 1
%o A048109         while kmax-kmin > 1:
%o A048109             kmid = kmax+kmin>>1
%o A048109             if f(kmid) <= kmid:
%o A048109                 kmax = kmid
%o A048109             else:
%o A048109                 kmin = kmid
%o A048109         return kmax
%o A048109     def g(x): return sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
%o A048109     def f(x): return int(n+x+sum(sum(-g(x//p**j) if j&1 else g(x//p**j) for j in range(3,x.bit_length())) for p in primerange(isqrt(x)+1)))
%o A048109     return bisection(f,n,n) # _Chai Wah Wu_, Feb 24 2025
%Y A048109 Cf. A000005, A001221, A034444, A036537, A048106, A048107.
%K A048109 nonn
%O A048109 1,1
%A A048109 _Labos Elemer_
%E A048109 New name based on comment by _Ivan Neretin_, Jun 19 2015