A294092 Numbers k == 119 (mod 120) such that 2^((k-1)/2), 3^((k-1)/2) and 5^((k-1)/2) are congruent to 1 (mod k).

239, 359, 479, 599, 719, 839, 1319, 1439, 1559, 2039, 2399, 2879, 2999, 3119, 3359, 3719, 4079, 4679, 4799, 4919, 5039, 5279, 5399, 5519, 5639, 5879, 6359, 6599, 6719, 6959, 7079, 7559, 7919, 8039, 8999, 9239, 9479, 9719, 9839, 10079, 10559, 10799, 11159, 11279

Offset

1,1

Comments

So far no composite numbers have been found in this sequence. There are no pseudoprimes up to 2^64 in this sequence, so a composite term in this sequence has to exceed 18446744066047760377.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

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

Mathematica

k = 119; lst = {}; While[k < 12000, If[ PowerMod[{2, 3, 5}, (k - 1)/2, k] == {1, 1, 1}, AppendTo[lst, k]];  k += 120]; lst (* Robert G. Wilson v, Feb 11 2018 *)

Prog

(PARI) is(n) = n%120==119 && Mod(2, n)^((n-1)\2)==1 && Mod(3, n)^((n-1)\2)==1 && Mod(5, n)^((n-1)\2)==1
(Python)
A294092_list, k, m = [], 119, 59
while len(A294092_list) < 10000:
    if pow(2, m, k) == 1 and pow(3, m, k) == 1 and pow(5, m, k) == 1:
        A294092_list.append(k)
    k += 120
    m += 60 # Chai Wah Wu, Feb 09 2018
(GAP) Filtered([1..14000], n->n mod 120=119 and 2^((n-1)/2) mod n =1 and 3^((n-1)/2) mod n =1 and 5^((n-1)/2) mod n =1); # Muniru A Asiru, Feb 15 2018

Crossrefs

Cf. A001567.

Sequence in context: A289109 A247888 A243102 * A056086 A046012 A152952

Adjacent sequences: A294089 A294090 A294091 * A294093 A294094 A294095

Keyword

nonn

Author

Jonas Kaiser, Feb 09 2018

Extensions

More terms from Chai Wah Wu, Feb 10 2018

Status

approved