A367742 Lexicographically earliest infinite sequence of distinct positive numbers such that, for n>3, a(n) has a common factor with a(n-2) and n but not with a(n-1).

1, 2, 3, 4, 15, 8, 21, 10, 9, 5, 33, 20, 39, 14, 27, 16, 51, 22, 57, 26, 45, 28, 69, 32, 75, 34, 63, 38, 87, 40, 93, 44, 81, 46, 105, 52, 111, 50, 99, 25, 123, 35, 129, 55, 6, 115, 94, 135, 56, 65, 12, 13, 106, 117, 80, 91, 18, 203, 118, 145, 122, 155, 183, 62, 165, 58, 201, 64, 141, 68, 213

Offset: 1

Author

Scott R. Shannon, Nov 29 2023

Keywords

• nonn

Comments

This is a variation of the Yellowstone permutation A098550 with an additional restriction that each term a(n) must have a common factor with n. For the sequence to be infinite a(n) must be chosen so it does not have as factors all the prime factors of n+1. See the examples below.

Unlike A098550 the primes do not appear in their natural order, and in general are delayed in their appearance relative to similarly sized numbers. In the first 100000 terms the fixed points are 1, 2, 3, 4, 9, 14, 16, 74, 76, 86, 88, 207, 320, 322, 901; it is unknown if more exist. The sequence is conjectured to be a permutation of the positive numbers.

Examples

			a(5) = 15 as 15 shares a factor with a(3) = 3 and with n = 5, does not share a factor with a(4) = 4, and 15 does not have as factors all the prime factors of 5+1 = 6 = 2*3.
a(55) = 80 as 80 shares a factor with a(53) = 106 and with n = 55, does not share a factor with a(54) = 117, and 80 does not have as factors all the prime factors of 55+1 = 56 = 2^3*7. Note that 70 satisfies the first three criteria but not the last, so choosing a(55) = 70 would mean a(56) would not exist.
		

Links

• Scott R. Shannon, Table of n, a(n) for n = 1..10000
• Scott R. Shannon, Image of the first 100000 terms for a(n) <= 500000. The green line is a(n) = n.

Crossrefs

Cf. A367741, A098550, A336957, A064413, A027748, A368231.