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).

11, 19, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 227, 251, 283, 307, 331, 347, 379, 419, 443, 467, 491, 499, 523, 547, 563, 571, 587, 619, 643, 659, 683, 691, 739, 787, 811, 827, 859, 883, 907, 947, 971, 1019, 1051, 1091, 1123, 1163, 1171, 1187, 1259, 1283

Offset

1,1

Comments

It appears that A007520 is a subsequence. Up to 10^7 there are no composites in this sequence.

The first composite is a(17465859) = 1397357851; there are probably infinitely many. - Charles R Greathouse IV, Nov 12 2017

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

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 *)

Prog

(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

Crossrefs

Cf. A244626, A294717, A293394, A070179, A007520.

Sequence in context: A184328 A260271 A275797 * A201719 A107201 A189888

Adjacent sequences: A294990 A294991 A294992 * A294994 A294995 A294996

Keyword

nonn

Author

Jonas Kaiser, Nov 12 2017

Status

approved