1,1

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)

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]

d[ x ] = 2^(r[ x ]+1) or A000005[ x ]=2^(A(001221[ x ])+1)=2*A034444[ x ].

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}.

A000005, A001221, A034444, A036537, A048106, A048107.

Sequence in context: A052349 A029607 A060476 this_sequence A068781 A038524 A162829

Adjacent sequences: A048106 A048107 A048108 this_sequence A048110 A048111 A048112

nonn

Labos E. (labos(AT)ana.sote.hu)