A361321 Lexicographically earliest infinite sequence of distinct elements of A000469 such that, for n > 2, a(n) has a common factor with a(n-1) but not with a(n-2).

1, 6, 10, 35, 21, 33, 22, 14, 91, 39, 15, 55, 77, 42, 26, 65, 85, 34, 38, 57, 51, 119, 70, 30, 69, 161, 133, 95, 110, 46, 299, 143, 66, 58, 145, 105, 78, 62, 155, 115, 138, 74, 185, 165, 87, 203, 154, 82, 123, 93, 217, 182, 86, 129, 111, 259, 238, 94, 141, 159

Offset: 1

Author

Scott R. Shannon, Rémy Sigrist and N. J. A. Sloane, Mar 09 2023

Keywords

• nonn

Comments

This sequence is a variant of A360519 where we only consider nonprime squarefree numbers (A000469).

Theorem: a(1) = 1, a(2) = 6; thereafter, a(n) is the smallest nonprime squarefree number m not yet in the sequence such that

(i) gcd(m, a(n-1)) > 1,

(ii) gcd(m, a(n-2)) = 1, and

(iii) m does not divide a(n-1).

Conjecture: The sequence is a permutation of A000469.

Links

• Scott R. Shannon, Table of n, a(n) for n = 1..100000
• Scott R. Shannon, White on black graph of first 50000 terms [The green line is x = y]
• Scott R. Shannon, Image of the first 500000 terms in color. The terms with a lowest prime factor of 2,3,5,7,9,11,13,17,19,>=23 are colored white, red, orange, yellow, green, blue, indigo, violet, gray respectively.
• Rémy Sigrist, PARI program

Crossrefs

Cf. A000469, A336957, A360519.

Programs

PARI

See Links section.