A036260 -id:A036260 - cp's OEIS frontend

A036259 Numbers k such that the multiplicative order of 2 modulo k is odd.

1, 7, 23, 31, 47, 49, 71, 73, 79, 89, 103, 127, 151, 161, 167, 191, 199, 217, 223, 233, 239, 263, 271, 311, 329, 337, 343, 359, 367, 383, 431, 439, 463, 479, 487, 497, 503, 511, 529, 553, 599, 601, 607, 623, 631, 647, 713, 719, 721, 727, 743, 751

Offset: 1

David W. Wilson

nonn

Odd numbers k such that A007733(k) = A002326((k-1)/2) is odd.

Closed under multiplication. - Emmanuel Vantieghem, May 07 2025

			2^3 = 1 mod 7, 3 is odd, so 7 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002326, A007733, A036260, A053006.

Select[Range[1, 999, 2], OddQ[MultiplicativeOrder[2, #]]&] (* Jean-François Alcover, Dec 20 2017 *)

is(n)=n%2 && znorder(Mod(2,n))%2 \\ Charles R Greathouse IV, Jun 24 2015

from sympy import n_order
from itertools import count, islice
def A036259_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:n_order(2,n)&1,count(max(startvalue,1)|1,2))
A036259_list = list(islice(A036259_gen(),20)) # Chai Wah Wu, Feb 07 2023

A306833 a(1) = 3; a(n+1) is the smallest k > a(n) such that 2^(k-1) == 1 (mod a(n)).

Original entry on oeis.org

3, 5, 9, 13, 25, 41, 61, 121, 221, 241, 265, 313, 469, 529, 760

Offset: 1

Thomas Ordowski, Mar 12 2019

This sequence is finite, the last term a(15) = 760 is even.
Conjecture: for any initial term a(1), this recursion gives a finite sequence (ends with an even term).
Theorem: for odd a(n), a(n+1) is even if and only if ord_{a(n)}(2) is odd and (a(n) mod ord_{a(n)}(2)) is odd.
The set of penultimate terms of the sequences is {A036259} \ {A036260}.

Cf. A002326, A036259, A036260.

lista(nn) = {a = 3; print1(a, ", "); for (n=2, nn, k = a+1; while (Mod(2, a)^(k-1) != 1, k++); a = k; print1(a, ", "); if (!(a%2), break););} \\ Michel Marcus, Mar 24 2019

cp's OEIS Frontend

A036259 Numbers k such that the multiplicative order of 2 modulo k is odd.

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