A175625 Numbers k such that gcd(k, 6) = 1, 2^(k-1) == 1 (mod k), and 2^(k-3) == 1 (mod (k-1)/2).

7, 11, 23, 31, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 683, 719, 839, 863, 887, 983, 1019, 1123, 1187, 1283, 1291, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907, 2027, 2039, 2063, 2099, 2207, 2447, 2459, 2543

Offset: 1

Author

Alzhekeyev Ascar M, Jul 28 2010, Jul 30 2010

Keywords

• nonn

Comments

All composites in this sequence are 2-pseudoprimes, A001567. That subsequence begins with 536870911, 46912496118443, 192153584101141163, with no other composites below 2^64 (the first two were found by 'venco' from the dxdy.ru forum), and contains the terms of A303448 that are not multiples of 3. Correspondingly, composite terms include those of the form A007583(m) = (2^(2m+1) + 1)/3 for m in A303009. The only known composite member not of this form is a(1018243) = 536870911.

Intended as a pseudoprimality test; note that many primes do not pass the third condition either.

Conjecture: The prime values belong to A039787. - Bill McEachen, Dec 27 2023

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001567, A007583, A303009, A303448.

Cf. A039787.

Programs

Mathematica

Select[Array[(6 # + (-1)^# - 3)/2 &, 3000], And[PowerMod[2, (# - 1), #] == 1, PowerMod[2, (# - 3), (# - 1)/2] == 1] &] (* Michael De Vlieger, Dec 27 2023 *)

PARI

isA175625(n) = gcd(n,6)==1 && Mod(2,n)^(n-1)==1 && Mod(2,n\2)^(n-3)==1

Extensions

Partially edited by N. J. A. Sloane, Jul 29 2010

Entry rewritten by Charles R Greathouse IV, Aug 04 2010

Comment and b-file from Charles R Greathouse IV, Sep 06 2010

Edited by Max Alekseyev, May 28 2014, Apr 24 2018