%I
%S 0,1,1,3,1,2,1,2,3,1,4,2,1,3,3,5,1,4,3,1,3,3,3,3,2,1,1,1,3,8,3,2,1,6,
%T 1,2,3,3,3,5,1,5,1,2,1,7,4,2,1,2,4,1,5,3,4,4,1,5,3,1,6,6,2,1,2,7,3,4,
%U 1,3,4,6,3,3,3,4,6,3,5,5,1,6,1,3,3,4,5,1,1,2,6,4,3,4,3,2,6,1,8,3,6,4,5,1,4
%N Number of distinct values of A075860(j) when j runs through composite numbers between nth and (n+1)th primes. That is, the counts of different fixedpoints[=prime] reached by iteration of function A008472(=sum of prime factors) initiated with composite values between two consecutive primes.
%C This count is smaller than A001223[n]1 and albeit not frequently but it can be one even if primes of borders are not twin primes. Such primes are collected into A082883.
%F a(n) = Card(Union(A075860(x)); x=1+p(n), ..., 1+p(n+1)).
%e Between p(23)=83 and p(24)=89, the relevant fixed points are
%e {5,13,2,2,13}, i.e., four are distinct from the 5 values, a(24)=4;
%e between p(2033)=17707 and p(2034)=170713, the fixedpoint set is {5,5,5,5,5}, so a(2033)=1, so a(88)=1.
%t ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w1], {w, 1, lf[x]}] sopf[x_] := Apply[Plus, ba[x]] Table[Length[Union[Table[FixedPoint[sopf, w], {w, 1+Prime[n], Prime[n+1]1}]]], {n, 1, 1000}]
%Y Cf. A075860, A008472, A082087, A082088, A082880, A082081, A001223.
%K nonn
%O 1,4
%A _Labos Elemer_, Apr 16 2003
