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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.
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%I #16 Apr 04 2021 01:03:31

%S 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 19,20,21,37,22,211,24,26,213,371,753,3251,28,29,30,235,547,31,32,33,

%U 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 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 This is a permutation of the positive terms.

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%e 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 a(7) = 23 as the prime factors of a(6) = 6 are 2 and 3, which, concatenated in increasing order, give 23;

%e 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 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 Array[a,100] (* _Giorgos Kalogeropoulos_, Apr 02 2021 *)

%o (Python)

%o from sympy import primefactors

%o def aupton(terms):

%o alst, aset = [1, 2], {1, 2}

%o while len(alst) < terms:

%o an = int("".join(map(str, primefactors(alst[-1]))))

%o if an in aset:

%o an = 1

%o while an in aset: an += 1

%o alst.append(an); aset.add(an)

%o return alst[:terms]

%o print(aupton(100)) # _Michael S. Branicky_, Apr 02 2021

%Y 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 base,nonn

%O 1,2

%A _Eric Angelini_ and _Carole Dubois_, Apr 02 2021