OFFSET
1,1
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?
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
EXAMPLE
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.
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 *)
PROG
(Magma) sopfr:=func<n|&+[j[1]*j[2]: j in Factorization(n)]>; 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
CROSSREFS
KEYWORD
nonn,look
AUTHOR
David James Sycamore, Jan 01 2020
STATUS
approved