A082882 Number of distinct values of A075860(j) when j runs through composite numbers between n-th and (n+1)-th primes. That is, the counts of different fixed-points[=prime] reached by iteration of function A008472(=sum of prime factors) initiated with composite values between two consecutive primes.

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

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 fixed-point 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*w-1], {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 A352522

Adjacent sequences: A082879 A082880 A082881 * A082883 A082884 A082885

Keyword

nonn

Author

Labos Elemer, Apr 16 2003

Status

approved