cp's OEIS Frontend

A294993 Numbers n > 1 such that all of 2^(n-1), 3^(n-1), 5^(n-1), (2*n-1)*(2^((n-1)/2)), 4*ceiling((3/4)*n)-2, and (2^((n+1)/2) + floor(n/4)*2^((n+3)/2)) are congruent to 1 (mod n).

%I A294993 #16 Nov 16 2017 00:31:03
%S A294993 11,19,43,59,67,83,107,131,139,163,179,211,227,251,283,307,331,347,
%T A294993 379,419,443,467,491,499,523,547,563,571,587,619,643,659,683,691,739,
%U A294993 787,811,827,859,883,907,947,971,1019,1051,1091,1123,1163,1171,1187,1259,1283
%N A294993 Numbers n > 1 such that all of 2^(n-1), 3^(n-1), 5^(n-1), (2*n-1)*(2^((n-1)/2)), 4*ceiling((3/4)*n)-2, and (2^((n+1)/2) + floor(n/4)*2^((n+3)/2)) are congruent to 1 (mod n).
%C A294993 It appears that A007520 is a subsequence. Up to 10^7 there are no composites in this sequence.
%C A294993 The first composite is a(17465859) = 1397357851; there are probably infinitely many. - _Charles R Greathouse IV_, Nov 12 2017
%H A294993 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 A294993 Select[Range[2, 1300], Function[n, AllTrue[Join[Prime[Range@3]^(n - 1), {(2 n - 1) (2^((n - 1)/2)), 4 Ceiling[3 n/4] - 2, (2^((n + 1)/2) + Floor[n/4]*2^((n + 3)/2))}], Mod[#, n] == 1 &]]] (* _Michael De Vlieger_, Nov 15 2017 *)
%o A294993 (PARI) is(n) = n%2 && Mod(2, n)^(n-1)==1 && Mod(3, n)^(n-1)==1 && Mod(5, n)^(n-1)==1 && (2*n-1)*Mod(2, n)^((n-1)/2)== 1 && Mod(4*ceil((3/4)*n)-2, n)==1 && Mod(2, n)^((n+1)/2)+floor(n/4)*Mod(2, n)^((n+3)/2)==1
%Y A294993 Cf. A244626, A294717, A293394, A070179, A007520.
%K A294993 nonn
%O A294993 1,1
%A A294993 _Jonas Kaiser_, Nov 12 2017