A392329 - OEIS

%I #17 Jan 15 2026 07:45:06

%S 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,

%T 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,

%U 8,11,13,18,11,30,2,8,17,16,6,21,18,19,12,13,7,5,19,16,17,7,18,6

%N 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.

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

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

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

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

%H Paolo Xausa, <a href="/A392329/b392329.txt">Table of n, a(n) for n = 1..10000</a>

%e Irregular array n\ iterations gpf(6*n+1):

%e   1:  7, 43, 37, 223, 103, 619, 743, 13, so a(1) = 8.

%e   2:  13, so a(2) = 1.

%e   3:  19, 23, 139, 167, 59, 71, 61, 367, 2203, 13219, 547, 67, 31, 17, 103, 619, 743, 13, so a(3) = 18.

%e   4:  5, 31, 17, 103, 619, 743, 13, so a(4) = 7.

%e   5:  31, 17, 103, 619, 743, 13, so a(5) = 6.

%e   6:  37, 223, 103, 619, 743, 13, so a(6) = 6.

%e   7:  43, 37, 223, 103, 619, 743, 13, so a(7) = 7.

%e   8:  7, 43, 37, 223, 103, 619, 743, 13, so a(8) = 8.

%e   9:  11, 67, 31, 17, 103, 619, 743, 13, so a(9) = 8.

%e   10: 61, 367, 2203, 13219, 547, 67, 31, 17, 103, 619, 743, 13, so a(10) = 12.

%e   11: 67, 31, 17, 103, 619, 743, 13, so a(11) = 7.

%e   12: 73, 439, 31, 17, 103, 619, 743, 13, so a(12) = 8.

%e   13: already 13, so a(13) = 0.

%e   14: 17, 103, 619, 743, 13, so a(14) = 5.

%e   15: 13, so a(15) = 1.

%e   16: 97, 53, 29, 7, 43, 37, 223, 103, 619, 743, 13, so a(16) = 11.

%t A392329[n_] := Length[NestWhileList[FactorInteger[6*# + 1][[-1, 1]] &, n, FreeQ[{13, 47}, #] &]] - 1;

%t Array[A392329, 100] (* _Paolo Xausa_, Jan 15 2026 *)

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

%Y Cf. A006530, A392240.

%K nonn

%O 1,1

%A _Alain Rocchelli_, Jan 07 2026