A365259 Lexicographically earliest sequence of distinct positive numbers such that, for n > 2, a(n) shares a factor with a(n-1) and a(n+a(n)).

1, 2, 4, 6, 3, 9, 12, 15, 5, 10, 8, 14, 7, 35, 21, 18, 16, 20, 22, 28, 24, 26, 30, 25, 40, 32, 34, 17, 51, 27, 33, 11, 44, 36, 38, 42, 39, 45, 48, 46, 50, 52, 66, 54, 68, 56, 49, 70, 55, 60, 57, 19, 76, 58, 29, 87, 63, 72, 62, 31, 124, 64, 74, 78, 65, 13, 91, 77, 84, 69, 114, 75, 80, 82, 41, 123

Offset

1,2

Comments

The majority of terms lie just above the line a(n) = n, although a small number of terms are significantly larger due to having to share a factor with numerous previous terms, some of which are prime, e.g., a(6254) = 4501732178.

In the first 10000 terms the fixed points are 1, 2, 10, 65, 1261, 6527, although more likely exist. The sequences is conjectured to be a permutation of the positive integers. Unlike A064413 the primes do not occur in their natural order.

Links

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

Example

a(4) = 6 as 6 shares a factor with a(3) = 4 and also shares a factor with a(2) = 2 which is required as a(2+a(2)) = a(4) = 6.
a(8) = 15 as 15 shares a factor with a(7) = 12 and also shares a factor with a(5) = 3 which is required as a(5+a(5)) = a(8) = 15. Note that 8 shares a factor with 12 but it does not share a factor with 3, so a(8) cannot be 8. This is the first term to differ from A064413.

Crossrefs

Cf. A064413, A027748, A365453, A366021.

Sequence in context: A357777 A348086 A122280 * A356851 A291577 A354434

Adjacent sequences: A365256 A365257 A365258 * A365260 A365261 A365262

Keyword

nonn

Author

Scott R. Shannon, Sep 03 2023

Status

approved