A354687 a(1) = 1; for n > 1, a(n) is the smallest positive number that has not yet appeared that shares a factor with a(n-1) and the difference | a(n) - a(n-1) | is distinct from all previous differences.

1, 2, 4, 8, 14, 6, 3, 12, 22, 10, 5, 20, 34, 16, 32, 52, 13, 26, 48, 15, 36, 9, 33, 44, 18, 46, 23, 69, 21, 28, 58, 24, 56, 7, 42, 78, 27, 72, 30, 55, 11, 66, 104, 38, 19, 76, 116, 29, 87, 141, 39, 91, 35, 85, 17, 102, 40, 100, 25, 90, 153, 45, 114, 50, 120, 192, 51, 68, 142, 54, 130, 208, 60

Offset

1,2

Comments

The terms are concentrated along many different lines, although three lines contain a higher concentration of terms than the others; these are similar to the three lines seen in A064413. See the linked image. The primes do not occur in their natural order, and unlike A064413, the terms proceeding and following a prime term can be high multiples of the prime.

In the first 200000 terms the fixed points are 1,2,6,10,68. It is plausible no more exist although this is unknown. The sequence is conjectured to be a permutation of the positive integers.

See A354721 for the differences between terms.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Annotated log log scatterplot of a(n), n = 1..2^14, showing records in red and local minima in blue, highlighting primes in green and fixed points in gold.

Scott R. Shannon, Image of the first 100000 terms. The green line is y = n.

Example

a(4) = 8 as a(3) = 4, and 8 is the smallest unused number that shares a factor with 4 and whose difference from the previous term,| 8 - 4 | = 4, has not appeared. Note 6 shares a factor with 4 but | 6 - 4 | = 2, and a difference of 2 has already occurred between as | a(3) - a(2) |, so 6 cannot be chosen.

Mathematica

nn = 120; c[_] = d[_] = 0; a[1] = c[1] = 1; a[2] = c[2] = j = 2; u = 3; Do[Set[k, u]; While[Nand[c[k] == 0, d[Abs[k - j]] == 0, ! CoprimeQ[j, k]], k++]; Set[{a[i], c[k], d[Abs[k - j]]}, {k, i, i}]; j = k; If[k == u, While[c[u] > 0, u++]], {i, 3, nn}]; Array[a, nn] (* Michael De Vlieger, Jun 04 2022 *)

Prog

(Python)
from math import gcd
from sympy import isprime, nextprime
from itertools import count, islice
def agen(): # generator of terms
    aset, diffset, an, mink = {1, 2}, {1}, 2, 3
    yield from [1, 2]
    for n in count(3):
        k = mink
        while k in aset or abs(an-k) in diffset or gcd(an, k) == 1: k += 1
        aset.add(k); diffset.add(abs(k-an)); an = k; yield an
        while mink in aset: mink += 1
print(list(islice(agen(), 73))) # Michael S. Branicky, Jun 04 2022

Crossrefs

Cf. A354721, A064413, A354688, A354731, A354087, A352763.

Sequence in context: A026665 A369704 A174540 * A253766 A368490 A076380

Adjacent sequences: A354684 A354685 A354686 * A354688 A354689 A354690

Keyword

nonn

Author

Scott R. Shannon, Jun 03 2022

Status

approved