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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). 0
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified June 21 08:18 EDT 2021. Contains 345358 sequences. (Running on oeis4.)