cp's OEIS Frontend

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).

%I A300101 #28 Nov 09 2023 08:52:44
%S A300101 34,14,23,46,77,48,68,186,44,75,47,117,112,273,19,312,390,10,221,160,
%T A300101 106,45,342,42,157,64,229,237,699,345,714,352,348,668,195,285,575,487,
%U A300101 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
%N 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).
%C A300101 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?
%C A300101 There is no value smaller than 9 for all the pseudoprimes below 2^64. - _Amiram Eldar_, Nov 09 2023
%H A300101 Amiram Eldar, <a href="/A300101/b300101.txt">Table of n, a(n) for n = 1..10000</a>
%H A300101 Jonas Kaiser, <a href="https://arxiv.org/abs/1608.00862">On the relationship between the Collatz conjecture and Mersenne prime numbers</a>, arXiv:1608.00862 [math.GM], 2016.
%F A300101 a(n) = (A001567(n) - 1) / A306413(n). - _Jianing Song_, Dec 12 2021
%t A300101 ((# - 1)/MultiplicativeOrder[2, #]) & /@ Select[Range[10^5], CompositeQ[#] && PowerMod[2, # - 1, #] == 1 &] (* _Amiram Eldar_, Nov 09 2023 *)
%o A300101 (PARI) is_A001567(n)={Mod(2, n)^n==2 & !isprime(n) & n>1};
%o A300101 lista(nn) = {for (n=1, nn, if (is_A001567(n), print1((n-1)/znorder(Mod(2, n)), ", ");););} \\ _Michel Marcus_, Feb 25 2018
%Y A300101 Cf. A001917, A001567, A306413, A174590.
%K A300101 nonn
%O A300101 1,1
%A A300101 _Jonas Kaiser_, Feb 24 2018