

A295835


Numbers k == 3 (mod 4) such that 2^((k1)/2), 3^((k1)/2) and 5^((k1)/2) are congruent to 1 (mod k).


1



71, 191, 239, 311, 359, 431, 479, 599, 719, 839, 911, 1031, 1151, 1319, 1439, 1511, 1559, 1871, 2039, 2111, 2351, 2399, 2591, 2711, 2879, 2999, 3119, 3191, 3359, 3671, 3719, 3911, 4079, 4271, 4391, 4679, 4751, 4799, 4871, 4919, 5039, 5231, 5279, 5351, 5399, 5471
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OFFSET

1,1


COMMENTS

There are very few composite numbers in this sequence: The probability of catching a pseudoprime number (A001567) with this definition is estimated at 1 in 263 billion.
Composite numbers in the sequence include the Carmichael numbers 131314855918751, 23282264781147191, 70122000249565031, 104782993259720471, 583701149409931151, 870012810301712351.  Robert Israel, Nov 28 2017
With the exception of the pseudoprimes, it seems that this is a subsequence of A139035. Primes of this form (A139035) have two special properties. 1. There exists a smallest m of the form m = (A139035  1)/2 such that 2^m == 1 (mod A139035). 2. m is odd. The core of this definition is based on these two properties. The term 2^((k1)/2) == 1 (mod n) is based on the first property, while the term k == 3 (mod 4) is based on the second property. The terms 3^((k1)/2) == 1 (mod n) and 5^((k1)/2) == 1 (mod n) I just tried freely to Fermat.
Prime terms are congruent to 71 or 119 modulo 120.  Jianing Song, Nov 22 2018


LINKS

Robert Israel, 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.


MAPLE

filter:= proc(n) [2&^((n1)/2), 3&^((n1)/2), 5&^((n1)/2)] mod n = [1, 1, 1] end proc:
select(filter, [seq(i, i=3..10000, 4)]); # Robert Israel, Nov 28 2017, corrected Feb 26 2018


MATHEMATICA

fQ[n_] := PowerMod[{2, 3, 5}, (n  1)/2, n] == {1, 1, 1}; Select[3 + 4Range@ 1500, fQ] (* Michael De Vlieger, Nov 28 2017 and slightly modified by Robert G. Wilson v, Feb 26 2018 based on the renaming *)


PROG

(PARI) is(n) = n%4==3 && Mod(2, n)^(n\2)==1 && Mod(3, n)^(n\2)==1 && Mod(5, n)^(n\2)==1 && Mod(2, n)^logint(n+1, 2)!=1 \\ Charles R Greathouse IV, Nov 28 2017


CROSSREFS

Cf. A001567, A293394, A294717, A294993, A294919, A294912.
Sequence in context: A331008 A068364 A142612 * A139991 A140007 A023107
Adjacent sequences: A295832 A295833 A295834 * A295836 A295837 A295838


KEYWORD

nonn


AUTHOR

Jonas Kaiser, Nov 28 2017


EXTENSIONS

Definition corrected by Jonas Kaiser, Feb 05 2018


STATUS

approved



