A300101 a(n) = (pp-1)/x, where pp = A001567(n) and x = ord(2,pp), the smallest positive integer such that 2^x == 1 (mod pp).

34, 14, 23, 46, 77, 48, 68, 186, 44, 75, 47, 117, 112, 273, 19, 312, 390, 10, 221, 160, 106, 45, 342, 42, 157, 64, 229, 237, 699, 345, 714, 352, 348, 668, 195, 285, 575, 487, 56, 163, 502, 9, 357, 439, 310, 296, 208, 803, 151, 684, 217, 2038, 324, 315, 1666, 344, 1973, 319, 607, 2763, 62, 1777, 1122, 1360, 1135, 2603

Offset: 1

Author

Jonas Kaiser, Feb 24 2018

Keywords

• nonn

Comments

For primes, this definition has a clear distribution over the natural numbers (see A001917), whereas there is no such distribution for pseudoprimes. Among the first 10^6 pseudoprimes of this sequence, the smallest number is 9. Are there any numbers in this sequence which are smaller than 9?

There is no value smaller than 9 for all the pseudoprimes below 2^64. - Amiram Eldar, Nov 09 2023

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Jonas Kaiser, On the relationship between the Collatz conjecture and Mersenne prime numbers, arXiv:1608.00862 [math.GM], 2016.

Crossrefs

Cf. A001917, A001567, A306413, A174590.

Programs

Mathematica

((# - 1)/MultiplicativeOrder[2, #]) & /@ Select[Range[10^5], CompositeQ[#] && PowerMod[2, # - 1, #] == 1 &] (* Amiram Eldar, Nov 09 2023 *)

PARI

is_A001567(n)={Mod(2, n)^n==2 & !isprime(n) & n>1};
lista(nn) = {for (n=1, nn, if (is_A001567(n), print1((n-1)/znorder(Mod(2, n)), ", ");););} \\ Michel Marcus, Feb 25 2018

Formula

a(n) = (A001567(n) - 1) / A306413(n). - Jianing Song, Dec 12 2021