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A029912 Start with n; repeatedly sum prime factors (counted with multiplicity) and add 1, until reach 1, 6 or a prime.

%I A029912 #11 Aug 21 2022 11:47:31
%S A029912 1,3,5,5,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,11,7,7,7,11,7,7,7,11,11,11,
%T A029912 11,7,7,13,11,7,7,17,7,13,13,7,7,7,7,7,7,7,13,11,7,7,7,17,7,23,11,13,
%U A029912 13,7,7,7,13,19,17,7,7,7,7,13,13,7,7,7,7,19,19,7,7,13,7,7,7,23,7
%N A029912 Start with n; repeatedly sum prime factors (counted with multiplicity) and add 1, until reach 1, 6 or a prime.
%C A029912 If p is in A023200 then a(3*p) = p+4.  It appears that all n > 35 such that a(n) > n/3 are 3*p for p in A023200. - _Robert Israel_, Dec 18 2019
%H A029912 Robert Israel, <a href="/A029912/b029912.txt">Table of n, a(n) for n = 1..10000</a>
%e A029912 20 -> 2+2+5+1 = 10 -> 2+5+1 = 8 -> 2+2+2+1 = 7 so a(20)=7.
%p A029912 f:= proc(n) option remember;
%p A029912   local v;
%p A029912   v:= add(t[1]*t[2],t=ifactors(n)[2])+1;
%p A029912   if v = 1 or v = 6 or isprime(v) then return v fi;
%p A029912   procname(v)
%p A029912 end proc:
%p A029912 map(f, [$1..100]); # _Robert Israel_, Dec 18 2019
%t A029912 a[n_] := a[n] = If[n==1, 1, Module[{v}, v = Sum[t[[1]]*t[[2]], {t, FactorInteger[n]}]+1; If[v==1 || v==6 || PrimeQ[v], Return[v]]; a[v]]];
%t A029912 a /@ Range[100] (* _Jean-François Alcover_, Aug 21 2022, after _Robert Israel_ *)
%K A029912 nonn
%O A029912 1,2
%A A029912 _Dann Toliver_