A373546 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number that shares a factor with a(n-1) if a(n-1) is even otherwise is coprime to a(n-1) if a(n-1) is odd.

1, 2, 4, 6, 3, 5, 7, 8, 10, 12, 9, 11, 13, 14, 16, 18, 15, 17, 19, 20, 22, 24, 21, 23, 25, 26, 28, 30, 27, 29, 31, 32, 34, 36, 33, 35, 37, 38, 40, 42, 39, 41, 43, 44, 46, 48, 45, 47, 49, 50, 52, 54, 51, 53, 55, 56, 58, 60, 57, 59, 61, 62, 64, 66, 63, 65, 67, 68, 70, 72, 69, 71, 73, 74, 76, 78, 75

Offset: 1

Author

Scott R. Shannon, Jun 09 2024

Keywords

• nonn

Comments

The sequence forms a repetitive pattern of three even numbers separated by 2 followed by three odd numbers separated by 2, the first odd number being 3 less than the third even number. One third of all terms are fixed points, and the primes appear in their natural order.

First differences repeat {1, 2, 2, -3, 2, 2}. - Michael De Vlieger, Jun 11 2024

Examples

			a(9) = 10 as a(8) = 8 is an even number and 10 is the smallest unused positive number to share a factor with 8.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Labeled scatterplot of a(n) n = 1..72, with red indicating prime, gold indicating odd, and dark blue indicating even terms.

Crossrefs

Cf. A373545, A064413, A005843, A005408.

Programs

Mathematica

kk = 2; nn = 120; c[_] := False; Array[Set[{a[#], c[#]}, {#, True}] &, kk];
j = a[kk]; u = kk + 1;
Do[If[EvenQ[j],
   If[PrimePowerQ[j],
     p = FactorInteger[j][[1, 1]];
       k = #1 + Boole[#2 > 0] & @@ QuotientRemainder[u, p];
       While[c[k  p], k++]; k *= p,
     k = u; While[Or[c[k], CoprimeQ[j, k]], k++]],
   k = u; While[Or[c[k], ! CoprimeQ[j, k]], k++] ];
  Set[{a[n], c[k], j}, {k, True, k}];
  If[k == u, While[c[u], u++]], {n, kk + 1, nn}];
Array[a, nn] (* Michael De Vlieger, Jun 11 2024 *)