A376288 Lexicographically earliest infinite sequence of distinct positive integers such that, for n > 2, a(n) shares a factor with a(n-2)+a(n-1) but not with |a(n-2)-a(n-1)|.

1, 5, 9, 21, 25, 23, 3, 39, 7, 69, 19, 11, 15, 13, 35, 27, 31, 29, 33, 93, 49, 71, 45, 87, 55, 213, 67, 63, 75, 115, 57, 43, 65, 51, 145, 77, 37, 81, 59, 85, 99, 207, 17, 91, 123, 107, 95, 101, 119, 121, 105, 113, 109, 111, 125, 177, 151, 41, 117, 79, 147, 565, 89, 129, 327, 133, 135, 201, 161

Offset

1,2

Comments

To ensure the sequence is infinite a(n) must be chosen so that a(n-1)+a(n) has one or more prime factors that are not factors of |a(n-1)-a(n)|. This criteria is first encountered when finding a(3). The smallest number that shares a factor with a(1)+a(2) = 6 while not sharing one with |a(1)-a(2)| = 4 is 3, but choosing 3 would halt the sequence since then a(4) would need to share a factor with 8 but not with 2.

The sequence starts with a(2) = 5 as it is naturally a sequence of odd numbers - starting with an even number will quickly result in two consecutive odd numbers, after which all terms must be odd so they do not share a factor with the difference of the previous two terms, which will always be even. Choosing a(2) = 3 however is not possible due to a similar reason to that given above.

The terms are slightly concentrated along many straight lines, with a higher concentration along a line slightly above a(n) = 2n. Terms below this line appear to be all prime values; see the attaching image of the first 50000 terms.

The primes do not occur in their natural order, and the only fixed point in the first 100000 terms is a(n) = 37; it is unknown if more exist.

Links

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

Scott R. Shannon, Image of the first 50000 terms. Prime values are shown in yellow. The green line is a(n) = n.

Formula

a(4) = 21 as a(2) + a(3) = 5 + 9 = 14 while |a(2) - a(3)| = |5 - 9| = 4, and 21 has a common factor with 14 while not sharing one with 4. Note that 7 also satisfies these criteria but it cannot be chosen else a(5) would not exist.

Crossrefs

Cf. A336957, A098550, A005408, A064413.

Sequence in context: A273388 A273449 A297362 * A175364 A244642 A211424

Adjacent sequences: A376285 A376286 A376287 * A376289 A376290 A376291

Keyword

nonn

Author

Scott R. Shannon, Sep 19 2024

Status

approved