A330807 a(1) = 2; a(n+1) = Sum_{k=1..n} {S(a(k)): S(a(k)) = S(a(n))}, where S is sopfr (A001414).

2, 2, 4, 4, 8, 6, 5, 10, 7, 14, 9, 12, 21, 10, 28, 11, 22, 13, 26, 15, 8, 18, 16, 24, 18, 32, 20, 27, 36, 30, 40, 33, 14, 45, 44, 30, 50, 12, 35, 24, 54, 55, 16, 40, 66, 32, 60, 36, 70, 28, 77, 18, 48, 88, 17, 34, 19, 38, 21, 80, 39, 48, 99, 51, 20, 63, 52, 68, 42, 48, 110, 36, 90, 65, 54, 121, 22, 78, 72, 60, 72, 84, 42, 96

Offset: 1

Author

David James Sycamore, Jan 01 2020

Keywords

• nonn
• look

Comments

a(n+1) is k(n)*sopfr(a(n)), where k(n) is the number of times (up to and including a(n)) that a term having the same sopfr as a(n) has occurred in the sequence so far.

The smallest possible initial term is a(1)=2, since 2 is the smallest number having a prime divisor. Not every integer appears; for example, 3 cannot occur unless chosen as a(1) (in which case 2 cannot appear).

The primes do not appear in their natural order (e.g., 31 precedes 29). If the lesser of twin primes p is a(k) and the greater twin has not already occurred, then a(k+2) is the greater twin. Whereas if a composite number appears, it may appear more than once, a prime > 2, if it appears, can appear once only. Open question: Do all primes (except for 3) eventually appear?

Examples

			a(2)=2 because it is the sopfr of a(1) and there are no prior terms which could contribute to the sum.
a(3)=2+2=4 because 2 has occurred twice already as sopf of prior terms.
a(7)=5 because a(6)=6 and has sopfr 5 which has not been seen before.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001414, A279818.

Programs

Magma

sopfr:=func; a:=[2]; for n in [2..90] do  Append(~a,&+[sopfr(a[k]):k in [1..n-1]|sopfr(a[k]) eq  sopfr(a[n-1])]); end for; a; // Marius A. Burtea, Jan 01 2020

Mathematica

s[1] = 0; s[n_] := Plus @@ Times @@@ FactorInteger[n]; a[1] = 2; a[n_] := a[n] = (s1 = s[a[n - 1]])*(1 + Sum[Boole[s[a[k]] == s1], {k, 1, n - 2}]); Array[a, 100] (* Amiram Eldar, Jan 01 2020 *)