A331603 a(1) = 1; for n > 1, if a(n-1) is composite then a(n) is the concatenation of all the prime factors in order of a(n-1), otherwise a(n) is the smallest number not yet appearing in the sequence.

1, 2, 3, 4, 22, 211, 5, 6, 23, 7, 8, 222, 2337, 31941, 33371313, 311123771, 7149317941, 22931219729, 112084656339, 3347911118189, 11613496501723, 97130517917327, 531832651281459, 3331113965338635107, 9, 33, 311, 10, 25, 55, 511, 773, 11, 12, 223, 13, 14, 27, 333, 3337, 4771, 13367, 15

Offset: 1

Author

Scott R. Shannon, Jan 21 2020

Keywords

• nonn
• base

Comments

Assuming that all numbers when replaced with the concatenation of their prime factors will eventually reach a prime (see A037274), this sequence will contain all positive integers. a(158) = 49 which currently has no known 'home prime' in the iterative sequence of prime factor replacements; see A056938.

Examples

			a(5) = 22 as a(4) = 4 which has a factorization 4 = 2*2, so the concatenation of factors is '22'.
a(7) = 5 as a(6) = 211 which is prime, and 5 is the smallest number not yet appearing in the sequence.
a(14) = 31941 as a(13) = 2337 which has a factorization 2337 = 3*19*41, so the concatenation of factors is '31941'.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..200
• Eric Weisstein's World of Mathematics, Home Prime.

Crossrefs

Cf. A000040, A027746, A037274, A037271, A006919, A056938.

Programs

Mathematica

nn = 43; c[] = 0; a[1] = c[1] = u = 1; While[c[u] > 0, u++]; Do[If[CompositeQ[#], k = FromDigits@ Flatten@ Map[IntegerDigits[#] &, ConstantArray[##] & @@@ FactorInteger[#]], k = u] &@ a[i - 1]; Set[{a[i], c[k]}, {k, i}]; If[k == u, While[c[u] > 0, u++]], {i, 2, nn}]; Array[a, nn] (* _Michael De Vlieger, Apr 12 2022 *)