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

1,1

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.

Formula

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A001917, A001567, A306413, A174590.

Sequence in context: A350934 A071788 A220133 * A298258 A298929 A118497

Adjacent sequences: A300098 A300099 A300100 * A300102 A300103 A300104

Keyword

nonn

Author

Jonas Kaiser, Feb 24 2018

Status

approved