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%I #21 Apr 13 2022 01:13:21
%S 1,2,3,4,22,211,5,6,23,7,8,222,2337,31941,33371313,311123771,
%T 7149317941,22931219729,112084656339,3347911118189,11613496501723,
%U 97130517917327,531832651281459,3331113965338635107,9,33,311,10,25,55,511,773,11,12,223,13,14,27,333,3337,4771,13367,15
%N 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.
%C 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.
%H Michael De Vlieger, <a href="/A331603/b331603.txt">Table of n, a(n) for n = 1..200</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HomePrime.html">Home Prime</a>.
%e a(5) = 22 as a(4) = 4 which has a factorization 4 = 2*2, so the concatenation of factors is '22'.
%e a(7) = 5 as a(6) = 211 which is prime, and 5 is the smallest number not yet appearing in the sequence.
%e a(14) = 31941 as a(13) = 2337 which has a factorization 2337 = 3*19*41, so the concatenation of factors is '31941'.
%t 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 *)
%Y Cf. A000040, A027746, A037274, A037271, A006919, A056938.
%K nonn,base
%O 1,2
%A _Scott R. Shannon_, Jan 21 2020
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