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

Offset: 1

Author

Jonas Kaiser, Nov 07 2017

Keywords

• nonn

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.

Crossrefs

Cf. A001133, A244626.

Programs

Mathematica

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

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