cp's OEIS Frontend

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

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