A392329 a(n) is the number of iterations x->gpf(6*x+1) starting at n until the value 13 or 47 is reached; a(n) = -1 if neither of these two values is reached.

8, 1, 18, 7, 6, 6, 7, 8, 8, 12, 7, 8, 0, 5, 1, 11, 4, 14, 17, 8, 15, 18, 16, 9, 6, 20, 6, 1, 8, 14, 5, 13, 21, 19, 15, 6, 5, 9, 1, 18, 18, 17, 6, 10, 19, 22, 0, 5, 14, 7, 12, 21, 9, 1, 9, 6, 8, 11, 13, 18, 11, 30, 2, 8, 17, 16, 6, 21, 18, 19, 12, 13, 7, 5, 19, 16, 17, 7, 18, 6

Offset

1,1

Comments

Conjecture: a(n) is never equal to -1.

For any n, the iteration of gpf(6*n+1) results in a cycle with a smallest term of 13 or 47.

Cycle 13: 79, 19, 23, 139, 167, 59, 71, 61, 367, 2203, 13219, 547, 67, 31, 17, 103, 619, 743, 13. (19 terms)

Cycle 47: 283, 1699, 2039, 2447, 14683, 8009, 1373, 107, 643, 227, 47. (11 terms)

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

Example

Irregular array n\ iterations gpf(6*n+1):
  1:  7, 43, 37, 223, 103, 619, 743, 13, so a(1) = 8.
  2:  13, so a(2) = 1.
  3:  19, 23, 139, 167, 59, 71, 61, 367, 2203, 13219, 547, 67, 31, 17, 103, 619, 743, 13, so a(3) = 18.
  4:  5, 31, 17, 103, 619, 743, 13, so a(4) = 7.
  5:  31, 17, 103, 619, 743, 13, so a(5) = 6.
  6:  37, 223, 103, 619, 743, 13, so a(6) = 6.
  7:  43, 37, 223, 103, 619, 743, 13, so a(7) = 7.
  8:  7, 43, 37, 223, 103, 619, 743, 13, so a(8) = 8.
  9:  11, 67, 31, 17, 103, 619, 743, 13, so a(9) = 8.
  10: 61, 367, 2203, 13219, 547, 67, 31, 17, 103, 619, 743, 13, so a(10) = 12.
  11: 67, 31, 17, 103, 619, 743, 13, so a(11) = 7.
  12: 73, 439, 31, 17, 103, 619, 743, 13, so a(12) = 8.
  13: already 13, so a(13) = 0.
  14: 17, 103, 619, 743, 13, so a(14) = 5.
  15: 13, so a(15) = 1.
  16: 97, 53, 29, 7, 43, 37, 223, 103, 619, 743, 13, so a(16) = 11.

Mathematica

A392329[n_] := Length[NestWhileList[FactorInteger[6*# + 1][[-1, 1]] &, n, FreeQ[{13, 47}, #] &]] - 1;
Array[A392329, 100] (* Paolo Xausa, Jan 15 2026 *)

Prog

(PARI) a(n) = my(nb=0); until(n==13 || n==47, nb++; n=vecmax(factor(6*n+1)[, 1])); nb;

Crossrefs

Cf. A006530, A392240.

Sequence in context: A359628 A369404 A209242 * A103884 A103883 A379447

Adjacent sequences: A392326 A392327 A392328 * A392330 A392331 A392332

Keyword

nonn

Author

Alain Rocchelli, Jan 07 2026

Status

approved