A352793 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not appeared that shares a single prime factor with a(n-1) and that prime has exponent 1 in the prime factorization of both a(n) and a(n-1).

1, 2, 6, 3, 12, 15, 5, 10, 14, 7, 21, 24, 33, 11, 22, 18, 26, 13, 39, 30, 34, 17, 51, 42, 35, 20, 45, 40, 55, 44, 77, 28, 63, 56, 91, 52, 65, 60, 57, 19, 38, 46, 23, 69, 48, 75, 66, 50, 54, 58, 29, 87, 78, 62, 31, 93, 84, 111, 37, 74, 70, 82, 41, 123, 96, 105, 80, 85, 68, 119, 102, 86, 43, 129

Offset: 1

Author

Scott R. Shannon, Apr 03 2022

Keywords

• nonn

Comments

No powerful number, see A001694, can be a term as the shared prime factor in both a(n) and a(n-1) must have exponent 1. In the first 200000 terms the fixed points are 1, 2, 944, 1700. It is possible more exist although this is unknown.

For the terms studied all primes appear in their natural order. Similarly to the EKG sequence, see A064413, any time a prime appears it is preceded by a multiple of the same prime. However, unlike the EKG sequence in which preceding term is always twice the prime, in this sequence at least four times the preceding term is three times the prime. This occurs at a(7) = 5, a(14) = 11, a(40) = 19, a(59) = 37. In the first 200000 terms no such further occurrences appear and it is unknown whether more exist.

Examples

			a(3) = 6 = 2*3 as a(2) = 2 and 6 is the smallest unused number which has a single 2 in its prime factorization. Note 4 = 2^2 so is not considered.
a(6) = 15 = 3*5 as a(5) = 12 = 2^2*3 and 15 is the smallest unused number which has a single 3 and contains no 2 in its prime factorization.
		

Links

• Scott R. Shannon, Table of n, a(n) for n = 1..10000
• Scott R. Shannon, Colored image of the first 200000 terms. The color corresponds to the minimum prime factor dividing the term - white, red, orange, yellow, green, blue, indigo, violet having minimum prime factor 2, 3, 5, 7, 11, 13, 17, 19 respectively, while terms with larger minimum prime factors are shown in grey. The solid green line is y = n.

Crossrefs

Cf. A001694, A064413, A000040, A005117.