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

1,1

Comments

It appears that A007521 is a subsequence.

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

Links

Table of n, a(n) for n=1..52.

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

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 *)

Prog

(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)

Crossrefs

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

Sequence in context: A100877 A261580 A007521 * A213050 A216822 A217466

Adjacent sequences: A294916 A294917 A294918 * A294920 A294921 A294922

Keyword

nonn

Author

_Jonas Kaiser_, Nov 10 2017

Extensions

More terms from _Alois P. Heinz_, Nov 10 2017

Status

approved