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A342999 a(n) is always followed by the concatenation of a(n)'s distinct prime factors in increasing order. If this concatenation is already in the sequence, a(n+1) is the smallest term not yet present.

%I A342999 #16 Apr 04 2021 01:03:31
%S A342999 1,2,3,4,5,6,23,7,8,9,10,25,11,12,13,14,27,15,35,57,319,1129,16,17,18,
%T A342999 19,20,21,37,22,211,24,26,213,371,753,3251,28,29,30,235,547,31,32,33,
%U A342999 311,34,217,731,1743,3783,31397,36,38,219,373,39,313,40,41,42,237,379,43,44,45,46,223,47
%N A342999 a(n) is always followed by the concatenation of a(n)'s distinct prime factors in increasing order. If this concatenation is already in the sequence, a(n+1) is the smallest term not yet present.
%C A342999 This is a permutation of the positive terms.
%H A342999 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e A342999 a(6) is not = 5, though the only prime factor of a(5) is precisely 5; but as 5 is already in the sequence we must take a(6) = 6, the smallest term not yet present in the sequence.
%e A342999 a(7) = 23 as the prime factors of a(6) = 6 are 2 and 3, which, concatenated in increasing order, give 23;
%e A342999 a(8) is not = 23, though the only prime factor of a(7) is precisely 23; but as 23 is already in the sequence we must take a(8) = 7, the smallest term not yet present in the sequence; etc.
%t A342999 a[1]=1;a[n_]:=a[n]=(g=FromDigits@Flatten[IntegerDigits@*First/@FactorInteger@a[n-1]];If[FreeQ[k=Array[a,n-1],g],g,Min@Complement[Range@Max[k+1],k]])
%t A342999 Array[a,100] (* _Giorgos Kalogeropoulos_, Apr 02 2021 *)
%o A342999 (Python)
%o A342999 from sympy import primefactors
%o A342999 def aupton(terms):
%o A342999   alst, aset = [1, 2], {1, 2}
%o A342999   while len(alst) < terms:
%o A342999     an = int("".join(map(str, primefactors(alst[-1]))))
%o A342999     if an in aset:
%o A342999       an = 1
%o A342999       while an in aset: an += 1
%o A342999     alst.append(an); aset.add(an)
%o A342999   return alst[:terms]
%o A342999 print(aupton(100)) # _Michael S. Branicky_, Apr 02 2021
%Y A342999 Cf. A084317 (concatenation of the prime factors of n, in increasing order), A037276 (replace n with the concatenation of its prime factors in increasing order).
%K A342999 base,nonn
%O A342999 1,2
%A A342999 _Eric Angelini_ and _Carole Dubois_, Apr 02 2021