A373902 a(1) = 2; for n > 1, a(n) is the smallest positive number that does not equal a(n-1), shares a factor with a(n-1), and has not appeared as an adjacent term to a(n-1) previously in the sequence.

2, 4, 6, 2, 8, 4, 10, 2, 12, 3, 6, 8, 10, 5, 15, 3, 9, 6, 10, 12, 4, 14, 2, 16, 4, 18, 2, 20, 4, 22, 2, 24, 3, 18, 6, 12, 8, 14, 6, 15, 9, 12, 14, 7, 21, 3, 27, 6, 16, 8, 18, 9, 21, 6, 20, 5, 25, 10, 14, 16, 10, 15, 12, 16, 18, 10, 20, 8, 22, 6, 24, 4, 26, 2, 28, 4, 30, 2, 32, 4, 34, 2, 36, 3

Offset

1,1

Comments

See A371618 for the indices where the primes first appear.

Links

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

Scott R. Shannon, Image of the first 250000 terms. Prime terms are shown in red.

Example

a(7) = 10 as a(6) = 4 and, although 2 and 6 share factors with 4, 2 and 4 form a previous adjacent pair, as do 4 and 6. This leaves 10 as the smallest number that shares a factor with 4 while 4 and 10 have not previously appeared as adjacent terms.

Mathematica

nn = 120; c[_] := {}; a[1] = j = 2; c[2] = {2};
Do[If[PrimePowerQ[j],
  (k = 1;
     While[Or[j == #  k, CoprimeQ[j, #  k], ! FreeQ[c[j], #  k]], k++];
     k *= #) &[FactorInteger[j][[1, 1]]],
   k = FactorInteger[j][[1, 1]];
     While[Or[j == k, CoprimeQ[j, k], ! FreeQ[c[j], k]], k++] ];
  Set[{a[n], c[j], c[k], j},
      {k, Union[c[j], {k}], Union[c[k], {j}], k}], {n, 2, nn}];
Array[a, nn] (* Michael De Vlieger, Jun 22 2024 *)

Prog

(Python)
from math import gcd
from itertools import count, islice
def agen(): # generator of terms
    an, adjacent = 2, {2: set()}
    while True:
        yield an
        A = adjacent[an]
        m = next(k for k in count(2) if k!=an and gcd(k, an)>1 and k not in A)
        adjacent[an].add(m)
        if m not in adjacent: adjacent[m] = set()
        adjacent[m].add(an)
        an = m
print(list(islice(agen(), 84))) # Michael S. Branicky, Jun 22 2024

Crossrefs

Cf. A371618, A064413, A027748, A364027.

Sequence in context: A278263 A236188 A352599 * A242593 A094752 A214061

Adjacent sequences: A373899 A373900 A373901 * A373903 A373904 A373905

Keyword

nonn,look

Author

Scott R. Shannon, Jun 22 2024

Status

approved