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

Author

Scott R. Shannon, Jun 22 2024

Keywords

• nonn
• look

Comments

See A371618 for the indices where the primes first appear.

Examples

			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.
		

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.

Crossrefs

Cf. A371618, A064413, A027748, A364027.

Programs

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 *)

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