A306413 -id:A306413 - 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).

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

Jonas Kaiser, Feb 24 2018

nonn

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

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.

Cf. A001917, A001567, A306413, A174590.

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

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

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

A367230 Base-2 Fermat pseudoprimes k such that the multiplicative order of 2 modulo k is odd.

Original entry on oeis.org

2047, 4681, 15841, 42799, 52633, 90751, 220729, 256999, 271951, 486737, 514447, 647089, 741751, 916327, 1082401, 1145257, 1730977, 1969417, 2205967, 2304167, 2748023, 2811271, 2953711, 2976487, 3567481, 4188889, 4469471, 4835209, 4863127, 5016191, 5049001, 5681809

Offset: 1

Amiram Eldar, Nov 11 2023

nonn

The corresponding sequence for primes is A014663.
These pseudoprimes seem to be relatively rare: among the 118968378 base-2 Fermat pseudoprimes below 2^64 only 6292535 are terms of this sequence.
These pseudoprimes appear in a theorem by Rotkiewicz and Makowski (1966) about pseudoprimes that are products of two Mersenne numbers (see A367229).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Andrzej Rotkiewicz and Andrzej Makowski, On Pseudoprime Numbers of the Form M_p M_t, Elemente der Mathematik, Vol. 21 (1966), pp. 133-134.

Intersection of A001567 and A036259.
A367231 is a subsequence.
Cf. A002326, A014663, A306413, A367229.

Select[2*Range[10^6] + 1, PowerMod[2, # - 1, #] == 1 && CompositeQ[#] && OddQ[MultiplicativeOrder[2, #]] &]

is(n) = n > 1 && n % 2 && Mod(2, n)^(n-1) == 1 && !isprime(n) && znorder(Mod(2, n)) % 2;

A350084 a(n) = ord(2,A006935(n)/2), where ord(k,m) is the multiplicative order of k modulo m.

Original entry on oeis.org

1, 261, 165, 275, 13425, 1485, 1305, 32085, 825, 3465, 2093, 3135, 495, 495, 261, 847, 9405, 552189, 198561, 261, 579261, 2475, 6237, 166725, 111111, 3393, 3565, 25245, 18585, 4437, 891891, 309455, 37125, 4833, 2301585, 14355, 11781, 3315, 915, 84975, 35259

Offset: 1

Jianing Song, Dec 12 2021

nonn

List of ord(2,k) where k ranges over the odd numbers such that 2^(2*k-1) == 1 (mod k).

			A006935(2) = 161038, so a(2) = ord(2,161038/2) = 261.
A006935(3) = 215326, so a(3) = ord(2,215326/2) = 165.

Amiram Eldar, Table of n, a(n) for n = 1..1319

Cf. A006935, A347906, A350083, A306413, A306414.

list(lim) = my(v=[],d); forstep(k=1, lim, 2, if((2*k-1)%(d=znorder(Mod(2,k)))==0, v=concat(v,d))); v \\ gives a(n) for A347906(n) <= lim

a(n) = ord(2,A347906(n)) = (A006935(n) - 1) / A350083(n).

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).

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A367230 Base-2 Fermat pseudoprimes k such that the multiplicative order of 2 modulo k is odd.

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A350084 a(n) = ord(2,A006935(n)/2), where ord(k,m) is the multiplicative order of k modulo m.

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