A294187 Numbers k == 77 (mod 120) such that (2*k-1)*2^((k-1)/2), (2*k-1)*3^((k-1)/2) and (2*k-1)*5^((k-1)/2) are congruent to 1 (mod k).

197, 317, 557, 677, 797, 1277, 1637, 1877, 1997, 2237, 2357, 2477, 2837, 2957, 3557, 3677, 3797, 3917, 4157, 4397, 4517, 4637, 4877, 5237, 5477, 5717, 6197, 6317, 6917, 7517, 7757, 7877, 8117, 8237, 8597, 8837, 9437, 9677, 10037

Offset

1,1

Comments

There are no composite numbers up to 2*10^17. The first composite term is 229467972529064957.

Links

Table of n, a(n) for n=1..39.

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

Maple

a:=k->`if`(k mod 120 = 77 and (2*k-1)*2^((k-1)/2) mod k = 1 and (2*k-1)*3^((k-1)/2) mod k = 1 and (2*k-1)*5^((k-1)/2) mod k = 1, k, NULL): seq(a(k), k=1..50); # Muniru A Asiru, Mar 11 2018

Mathematica

k = 77; lst = {}; While[k < 12000, If[Mod[(2k -1) PowerMod[{2, 3, 5}, (k -1)/2, k], k] == {1, 1, 1}, AppendTo[lst, k]]; k += 120]; lst (* Robert G. Wilson v, Feb 13 2018 *)

Prog

(PARI) is(n) = n%120==77 &&(2*n-1)* Mod(2, n)^((n-1)\2)==1 &&(2*n-1)* Mod(3, n)^((n-1)\2)==1 &&(2*n-1)* Mod(5, n)^((n-1)\2)==1 \\
(GAP) Filtered([1..11000], k->k mod 120 = 77 and (2*k-1)*2^((k-1)/2) mod k = 1 and (2*k-1)*3^((k-1)/2) mod k = 1 and (2*k-1)*5^((k-1)/2) mod k = 1); # Muniru A Asiru, Mar 11 2018

Crossrefs

Cf. A001567.

Sequence in context: A142121 A142274 A142624 * A158223 A142503 A142568

Adjacent sequences: A294184 A294185 A294186 * A294188 A294189 A294190

Keyword

nonn

Author

Jonas Kaiser, Feb 11 2018

Status

approved