A353026 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not appeared that is a multiple of the smallest prime factor with minimal exponent of a(n-1) (cf. A067695).

1, 2, 4, 6, 8, 10, 12, 3, 9, 15, 18, 14, 16, 20, 5, 25, 30, 22, 24, 21, 27, 33, 36, 26, 28, 7, 35, 40, 45, 50, 32, 34, 38, 42, 44, 11, 55, 60, 39, 48, 51, 54, 46, 52, 13, 65, 70, 56, 49, 63, 77, 84, 57, 66, 58, 62, 64, 68, 17, 85, 75, 69, 72, 78, 74, 76, 19, 95, 80, 90, 82, 86, 88, 99, 110

Offset: 1

Author

Scott R. Shannon, Apr 18 2022

Keywords

• nonn

Comments

The sequences is conjectured to be a permutation of the positive integers. In the first 500000 terms any time a prime p appears, where p>=5, the following term is 5p. It is unknown if this is true for all primes. In the same range the fixed points are 1, 2, 9, 39, 49, 1079, 4897, although it is possible more exist.

Examples

			a(4) = 6 as a(3) = 4 = 2*2 which has A067695(4) = 2 as the smallest prime factor with minimal exponent, and 6 is the smallest unused number that is a multiple of 2.
a(8) = 3 as a(7) = 12 = 2*2*3 which has A067695(12) = 3 as the smallest prime factor with minimal exponent, and 3 is the smallest unused number that is a multiple of 3.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Scott R. Shannon, Image of the first 100000 terms. The green line is y = n.

Crossrefs

Cf. A067695, A352793, A064413.

Programs

Mathematica

nn = 75, c[] = 0; Array[Set[{a[#], c[#]}, {#, #}] &, 2]; u = 3; Do[p = MinimalBy[FactorInteger@ a[i - 1], Last][[1, 1]]; k = u; While[Nand[c[k] == 0, Divisible[k, p]], k++]; Set[{a[i], c[k]}, {k, i}]; If[k == u, While[c[u] > 0, u++]], {i, Length[s] + 1, nn}]; Array[a, nn] (* _Michael De Vlieger, Apr 18 2022 *)