A294717 - OEIS

A294717

Numbers k such that 2^((k-1)/3) == 1 (mod k) and (2*k-1)*(2^((k-1)/6)) == 1 (mod k).

1, 43, 109, 157, 229, 277, 283, 307, 397, 499, 643, 691, 733, 739, 811, 997, 1021, 1051, 1069, 1093, 1459, 1579, 1597, 1627, 1699, 1723, 1789, 1933, 2179, 2203, 2251, 2341, 2347, 2731, 2749, 2917, 2971, 3061, 3163, 3181, 3229, 3259, 3277, 3331, 3373, 3541, 4027

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OFFSET

1,2

COMMENTS

Most of the elements of this sequence are prime. The "pseudoprimes" of these sequence are part of A244626.

LINKS

Charles R Greathouse IV, 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.

MATHEMATICA

Select[Range[1, 6001, 6], # == 1 || PowerMod[2, (#-1)/3, #] == 1 && Mod[-PowerMod[2, (#-1)/6, #], #] == 1&] (* Jean-François Alcover, Nov 18 2018 *)

PROG

(PARI) is(n)=n%6==1 && Mod(2, n)^(n\3)==1 && (2*n-1)*Mod(2, n)^(n\6)==1 \\ Charles R Greathouse IV, Nov 08 2017