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 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. 2
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified December 6 10:23 EST 2022. Contains 358630 sequences. (Running on oeis4.)