A374739 a(1) = 1, a(2) = 4; for n > 2, a(n) is the smallest unused positive number that shares a factor with a(n-1) while a(n)/gcd(a(n),a(n-1)) does not equal any previous term.

1, 4, 6, 9, 15, 10, 14, 16, 12, 8, 20, 22, 26, 34, 36, 21, 33, 39, 51, 54, 30, 18, 27, 45, 25, 35, 49, 77, 55, 65, 85, 95, 38, 46, 48, 28, 42, 57, 69, 72, 40, 24, 44, 52, 58, 62, 64, 56, 68, 74, 82, 86, 94, 100, 60, 50, 70, 91, 119, 133, 161, 115, 145, 87, 93, 96, 66, 78, 102, 106, 118, 122, 126

Offset

1,2

Comments

The sequence shows long runs of both even and odd terms; in the first 100000 terms the longest run of even terms is 979 while the longest run of odd terms is 3668. In the same range the vast majority of terms with a(n) > n are odd; only 914 even terms are above this line while 60073 odd terms are, the majority of the later having only two prime factors - see the linked image.

Unless the sequence starts with primes no other primes can appear in the sequence, hence is natural to start the sequence with a(1) = 1 and a(2) = 4.

The fixed points begin 1, 25, 548, 1617, 2763, 3897, 5253, although it is likely there are many more.

Links

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

Scott R. Shannon, Image of the first 100000 terms. Numbers with two, three, four, or five and more prime factors, counted with multiplicity, are show as yellow, green, blue and violet respectively. The white line is a(n) = n.

Example

a(3) = 6 as 6 shares a factor with a(2) = 4 and 6/gcd(6,4) = 3, and 3 does not equal any previous term.
a(10) = 8 as 8 shares a factor with a(9) = 12 and 8/gcd(8,12) = 2, and 2 does not equal any previous term.

Crossrefs

Cf. A064413, A003989, A112543.

Sequence in context: A179463 A086697 A372533 * A287568 A251728 A078443

Adjacent sequences: A374736 A374737 A374738 * A374740 A374741 A374742

Keyword

nonn,look

Author

Scott R. Shannon, Jul 18 2024

Status

approved