A391017 Starting values k of Collatz orbits that achieve a new minimum of Product_{j == 4 mod 6 in "3x+1" orbit of k} (j-1)/j.

1, 3, 7, 9, 25, 27, 33, 39, 57, 123, 219, 505, 519, 559, 745, 993, 2649, 3531, 12555, 35271, 117439, 141085, 156585, 167211, 594529, 626335, 668845, 792705, 891793, 1056939, 1879003, 2505337, 3340449, 4393983, 10415367, 21945135, 78027147, 194847855, 230930791

Offset

1,2

Comments

It is conjectured that A391016 is a lower bound of these products.

Links

Hugo Pfoertner, Table of n, a(n) for n = 1..60

Hugo Pfoertner, Table of n, a(n), A391018(n) and product, (2025).

Example

a(1) = 1: the empty product = 1.0;
a(2) = 3: the orbit of 3 is 3, 10, 5, 16, 8, 4, 2, 1. Terms == 4 mod 6 in this orbit are 10, 16 and 4. Prod_{j=[10,16,4]} (j-1)/j = (9/10) * (15/16) * (3/4) = 81/128 = 0.6328125 < 1.0.
a(3) = 7: the orbit of 7 is 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. Terms == 4 mod 6 in this orbit are 22, 34, 52, 40, 10, 16, 4. Prod_{j=[22,34,52,40,10,16,4]} (j-1)/j = (21/22) * (33/34) * (51/52) * (39/40) * (9/10) * (15/16) * (3/4) = 45927/81920 = 0.56063232421875 < a(2).

Prog

(PARI) a391017(upto) = my(smin=1); print1(1, ", "); forstep(n=1, upto, 2, my(k=n, s=1); while(k>1, if(k%6==4, s*=(k-1)/k); k=if(k%2, 3*k+1, k>>1)); if(s<smin, print1(n, ", "); smin=s))

Crossrefs

Cf. A006577, A391016, A391018.

Sequence in context: A099886 A118564 A098338 * A033958 A380347 A352016

Adjacent sequences: A391014 A391015 A391016 * A391018 A391019 A391020

Keyword

nonn

Author

Hugo Pfoertner and Markus Sigg, Nov 26 2025

Status

approved