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

1,2

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.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..200

Eric Weisstein's World of Mathematics, Home Prime.

Example

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'.

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 *)

Crossrefs

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

Sequence in context: A019075 A006709 A115884 * A171728 A010345 A233344

Adjacent sequences: A331600 A331601 A331602 * A331604 A331605 A331606

Keyword

nonn,base

Author

Scott R. Shannon, Jan 21 2020

Status

approved