A318387 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)).

6, 8, 15, 26, 69, 134, 393, 1556, 4659, 9314, 27933, 921327, 85680249, 171360494, 2227686253, 17821489976, 124750429783, 19336316610785, 4544034403522255, 3567067006764843005, 203322819385596050031, 25008706784428314148401, 825287323886134366896771, 91606892951360914725537141, 1923744751978579209236279751

Offset: 1

Author

David James Sycamore, Aug 25 2018

Keywords

• nonn

Comments

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).

Examples

			a(2) = 8, the smallest number whose sopfr is 6: A056240(8) = 6;
a(3) = 15, the smallest number whose sopfr is 8: A056240(8) = 15; etc.
		

Crossrefs

Cf. A001414, A048133, A002217, A056240.

Programs

Mathematica

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 *)

Formula

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

Extensions

a(13)-a(17) from Giovanni Resta, Aug 28 2018

Terms a(18) onward from Max Alekseyev, Sep 20 2024