Search: id:A048109
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%I A048109
%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 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^(alfa(1)+...+alfa(r)), 
               sigma_0(n) number of divisors of n (A000005), prime factorization 
               of n=p(1)^alfa(1) * ... * p(r)^alfa(r), alfa(1)+...+alfa(r) is sequence 
               (A086436). 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 A067295. [From Ctibor O. Zizka (c.zizka(AT)email.cz), Sep 
               21 2008]
%F A048109 d[ x ] = 2^(r[ x ]+1) or A000005[ x ]=2^(A(001221[ x ])+1)=2*A034444[ 
               x ].
%e A048109 n=88=2*2*2*11 has 8 divisors of which 4 is unitary divisors- because 
               of 2 distinct prime factors - and has 4 non unitary divisors: U={1,
               88,11,8} and NU = {2,44,4,22}.
%Y A048109 A000005, A001221, A034444, A036537, A048106, A048107.
%Y A048109 Sequence in context: A052349 A029607 A060476 this_sequence A068781 A038524 
               A162829
%Y A048109 Adjacent sequences: A048106 A048107 A048108 this_sequence A048110 A048111 
               A048112
%K A048109 nonn
%O A048109 1,1
%A A048109 Labos E. (labos(AT)ana.sote.hu)

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