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%I #30 Dec 27 2023 16:53:38
%S 7,11,23,31,47,59,83,107,167,179,227,263,347,359,383,467,479,503,563,
%T 587,683,719,839,863,887,983,1019,1123,1187,1283,1291,1307,1319,1367,
%U 1439,1487,1523,1619,1823,1907,2027,2039,2063,2099,2207,2447,2459,2543
%N Numbers k such that gcd(k, 6) = 1, 2^(k-1) == 1 (mod k), and 2^(k-3) == 1 (mod (k-1)/2).
%C 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.
%C Intended as a pseudoprimality test; note that many primes do not pass the third condition either.
%C Conjecture: The prime values belong to A039787. - _Bill McEachen_, Dec 27 2023
%H Charles R Greathouse IV, <a href="/A175625/b175625.txt">Table of n, a(n) for n = 1..10000</a>
%t Select[Array[(6 # + (-1)^# - 3)/2 &, 3000], And[PowerMod[2, (# - 1), #] == 1, PowerMod[2, (# - 3), (# - 1)/2] == 1] &] (* _Michael De Vlieger_, Dec 27 2023 *)
%o (PARI) isA175625(n) = gcd(n,6)==1 && Mod(2,n)^(n-1)==1 && Mod(2,n\2)^(n-3)==1
%Y Cf. A001567, A007583, A303009, A303448.
%Y Cf. A039787.
%K nonn
%O 1,1
%A _Alzhekeyev Ascar M_, Jul 28 2010, Jul 30 2010
%E Partially edited by _N. J. A. Sloane_, Jul 29 2010
%E Entry rewritten by _Charles R Greathouse IV_, Aug 04 2010
%E Comment and b-file from _Charles R Greathouse IV_, Sep 06 2010
%E Edited by _Max Alekseyev_, May 28 2014, Apr 24 2018
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