A365703 - OEIS

%I #15 Sep 16 2023 15:30:51

%S 1,2,12,3,13,113,1113,7,17,117,23,123,30,5,15,25,35,27,31,131,1131,29,

%T 129,32,20,21,33,11,110,22,24,26,28,37,137,1137,34,42,36,38,19,119,47,

%U 147,39,43,143,111,53,153,63,57,73,173,1173,83,183,61,161,67,167,1167,93,103,1030,45,50,51

%N a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared whose string value contains at least one prime factor of a(n-1).

%C This is a base-10 string version of the EKG sequence A064413. Any number that contains only nonprime digits, does not contain any substring that is prime, and is not itself a prime can never appear, e.g., 4, 6, 8, 9, 10, 14. The first number to contain only digits 4, 6, 8 or 9 that does appear is a(524) = 489, as it contains the substring 89 which is prime and is a factor of a(523) = 534.

%C The fixed points begin 15, 72, 248, 249, 670, 1055, 1056, 1377, although it is likely their total number is unbounded.

%H Scott R. Shannon, <a href="/A365703/b365703.txt">Table of n, a(n) for n = 1..10000</a>.

%e a(3) = 12 as a(2) = 2 and "12" is the smallest unused number to contain "2".

%e a(4) = 3 as a(3) = 12 = 2*3, and "3" is the smallest unused number to contain "3".

%e a(5) = 13 as a(4) = 3 and "13" is the smallest unused number to contain "3".

%e a(6) = 113 as a(5) = 13 and "113" is the smallest unused number to contain "13".

%t nn = 120; c[_] := False; f[x_] := FactorInteger[x][[All, 1]];

%t g[x_] := g[x] = Select[FromDigits /@ Union@ Rest@ Subsequences@ IntegerDigits[x], PrimeQ];

%t Array[Set[{a[#], c[#]}, {#, True}] &, 2]; j = 2; u = 3;

%t Do[Set[{j, k}, {f[j], u}];

%t  While[Or[c[k], # == {}, ! IntersectingQ[j, #]] &[g[k]], k++];

%t  Set[{a[n], c[k], j}, {k, True, k}];

%t  If[k == u, While[Or[c[u], g[u] == {}], u++]], {n, 3, nn}];

%t Array[a, nn] (* _Michael De Vlieger_, Sep 16 2023 *)

%Y Cf. A064413, A027748, A027746.

%K nonn,base

%O 1,2

%A _Scott R. Shannon_, Sep 16 2023