%I #20 Jan 20 2023 01:33:03

%S 6,8,15,26,69,134,393,1556,4659,9314,27933,921327,85680249,171360494,

%T 2227686253,17821489976,124750429783

%N Starting with a(1) = 6, a(n) is the smallest number whose sum of prime divisors (taken with multiplicity) is a(n-1). In other words, a(n) = A056240(a(n-1)).

%C Any nonzero number other than 4 or a prime could be chosen for a(1) so as to generate a nontrivial sequence (because A056240(r)=r for r=4 or a prime). In this sequence a(1) is set to 6 because it is the smallest composite number which is the sum of prime divisors of a greater number (8), and is therefore the smallest starting value for a non-stationary sequence of this kind. Let m = A056240(a(n-1)-q), where q is the greatest (prime or 4) < a(n-1)-1. Then a(n) = m*q, since sopfr(m*q) = sopf(m)+sopf(q) = a(n-1). Each term represents a step up (from the previous term) in the number of repeated iterations of sopfr required to reach a prime; a(n) >= A048133(n).

%F a(n) = A056240(a(n-1)); A002217(a(n)) = 1 + A002217(a(n - 1))

%e a(2) = 8, the smallest number whose sopfr is 6: A056240(8) = 6;

%e a(3) = 15, the smallest number whose sopfr is 8: A056240(8) = 15; etc.

%t With[{s = Array[Total@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger@ #] &,10^6]}, Nest[Append[#, First@ FirstPosition[s, #[[-1]] ]] &, {6}, 11]] (* _Michael De Vlieger_, Aug 25 2018 *)

%Y Cf. A001414, A048133, A002217, A056240.

%K nonn,more

%O 1,1

%A _David James Sycamore_, Aug 25 2018

%E a(13)-a(17) from _Giovanni Resta_, Aug 28 2018