

A082882


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.


1



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, 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, 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
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OFFSET

1,4


COMMENTS

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.


LINKS

Table of n, a(n) for n=1..105.


FORMULA

a(n) = Card(Union(A075860(x)); x=1+p(n), ..., 1+p(n+1)).


EXAMPLE

Between p(23)=83 and p(24)=89, the relevant fixed points are
{5,13,2,2,13}, i.e., four are distinct from the 5 values, a(24)=4;
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.


MATHEMATICA

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}]


CROSSREFS

Cf. A075860, A008472, A082087, A082088, A082880, A082081, A001223.
Sequence in context: A298421 A080131 A319956 * A188902 A324081 A256262
Adjacent sequences: A082879 A082880 A082881 * A082883 A082884 A082885


KEYWORD

nonn


AUTHOR

Labos Elemer, Apr 16 2003


STATUS

approved



