cp's OEIS Frontend

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).

%I A294919 #37 Feb 18 2019 09:55:06
%S A294919 5,13,29,37,53,61,101,109,149,157,173,181,197,229,269,277,293,317,349,
%T A294919 373,389,397,421,461,509,541,557,613,653,661,677,701,709,733,757,773,
%U A294919 797,821,829,853,877,941,997,1013,1021,1061,1069,1093,1109,1117,1181,1213
%N 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).
%C A294919 It appears that A007521 is a subsequence.
%C A294919 a(118) = 3277 = 29*113 is the first nonprime term.
%H A294919 Jonas Kaiser, <a href="https://arxiv.org/abs/1608.00862">On the relationship between the Collatz conjecture and Mersenne prime numbers</a>, arXiv:1608.00862 [math.GM], 2016.
%t A294919 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&];
%t A294919 Select[Range[1300], okQ] (* _Jean-François Alcover_, Feb 18 2019 *)
%o A294919 (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)
%Y A294919 Cf. A007521, A070179, A244626, A293394, A294717.
%K A294919 nonn
%O A294919 1,1
%A A294919 _Jonas Kaiser_, Nov 10 2017
%E A294919 More terms from _Alois P. Heinz_, Nov 10 2017