A387080 a(1)=1, a(2)=3; thereafter a(n) is either the greatest number k < a(n-1) not already used such that gcd(k, a(n-1)) > 1, or if no such k exists then a(n) is the smallest number k > a(n-1) not already used such that gcd(k, a(n-1)) > 1.

1, 3, 6, 4, 2, 8, 10, 5, 15, 12, 9, 18, 16, 14, 7, 21, 24, 22, 20, 25, 30, 28, 26, 13, 39, 36, 34, 32, 38, 19, 57, 54, 52, 50, 48, 46, 44, 42, 40, 35, 45, 33, 27, 51, 17, 68, 66, 64, 62, 60, 58, 56, 49, 63, 69, 23, 92, 90, 88, 86, 84, 82, 80, 78, 76, 74, 72, 70

Offset: 1

Author

Geoffrey Caveney and Michael De Vlieger, Aug 16 2025

Keywords

• nonn

Comments

This is a variant of A386482 that begins with 1,3 instead of 1,2.

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
• Michael De Vlieger, Log log scatterplot of a(n) in red and A386482(n) in blue, n = 1..2^20.
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^16, showing primes in red, proper prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, where the latter represents powerful numbers that are not prime powers.

Crossrefs

Cf. A386482.

Programs

Mathematica

Block[{c, j, k, m, p, r, nn},
  nn = 2^12; c[] := False; m[] := 1; j = 2; c[1] = c[2] = True; r = 1;
  {1}~Join~Monitor[Most@ Reap[Do[
    If[PrimePowerQ[j],
      Set[{p, k, m}, {#1, #1^(#2 - 1), #1^(#2 - 1)}] & @@
        FactorInteger[j][[1]]; While[And[c[k*p], k != 0], k--];
        If[k == 0, k = m; While[c[k*p], k++]]; k *= p,
      k = j - 1; While[And[Or[c[k], CoprimeQ[j, k]], k != 1], k--];
        If[k == 1, k += j; While[Or[c[k], CoprimeQ[j, k] ], k++] ] ];
    If[Mod[j, 2] == Mod[k, 2], r++, Sow[r]; r = 1];
    Set[{c[k], j}, {True, k}], {n, 3, nn}] ][[-1, 1]], n] ]