A365703 - OEIS

A365703

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).

1, 2, 12, 3, 13, 113, 1113, 7, 17, 117, 23, 123, 30, 5, 15, 25, 35, 27, 31, 131, 1131, 29, 129, 32, 20, 21, 33, 11, 110, 22, 24, 26, 28, 37, 137, 1137, 34, 42, 36, 38, 19, 119, 47, 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

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OFFSET

1,2

COMMENTS

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.

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

LINKS

Scott R. Shannon, Table of n, a(n) for n = 1..10000.

EXAMPLE

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

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

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

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

MATHEMATICA

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

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

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