A294919 Numbers n such that 2^(n-1), (2*n-1)*(2^((n-1)/2)), (4*ceiling((1/4)*n)-2), and (2^((n+1)/2) + floor((3/4)*n)*2^(((n+1)/2)+1)) are all congruent to 1 (mod n).

5, 13, 29, 37, 53, 61, 101, 109, 149, 157, 173, 181, 197, 229, 269, 277, 293, 317, 349, 373, 389, 397, 421, 461, 509, 541, 557, 613, 653, 661, 677, 701, 709, 733, 757, 773, 797, 821, 829, 853, 877, 941, 997, 1013, 1021, 1061, 1069, 1093, 1109, 1117, 1181, 1213

Offset: 1

Author

Jonas Kaiser, Nov 10 2017

Keywords

• nonn

Comments

It appears that A007521 is a subsequence.

a(118) = 3277 = 29*113 is the first nonprime term.

Links

• Jonas Kaiser, On the relationship between the Collatz conjecture and Mersenne prime numbers, arXiv:1608.00862 [math.GM], 2016.

Crossrefs

Cf. A007521, A070179, A244626, A293394, A294717.

Programs

Mathematica

okQ[n_] := AllTrue[{2^(n-1), (2*n-1)*(2^((n-1)/2)), (4*Ceiling@(n/4) - 2), (2^((n+1)/2) + Floor@((3/4)*n)*2^(((n+1)/2) + 1))}, Mod[#, n] == 1&];
Select[Range[1300], okQ] (* Jean-François Alcover, Feb 18 2019 *)

PARI

isok(n) = (n%2) && lift((Mod(2, n)^(n-1))==1)&&lift((Mod((2*n-1), n)*Mod(2, n)^((n-1)/2)) == 1)&&lift((Mod(((4*ceil((1/4)*n)-2)), n) )== 1)&&lift((Mod(2, n)^((n+1)/2) +Mod(floor((3/4)*n),n)*Mod(2, n)^(((n+1)/2)+1 ))== 1)

Extensions

More terms from Alois P. Heinz, Nov 10 2017