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A192297 Lesser of pseudo twin primes to base 2.

%I A192297 #58 Nov 10 2023 09:23:48
%S A192297 561,643,645,1103,1905,2465,2699,2819,4369,4371,4679,6599,10259,12799,
%T A192297 14489,16703,18719,19949,23001,25759,25761,29339,30119,31607,33151,
%U A192297 39863,41039,42797,49139,52631,55243,60701,62743,68099,72883,83663,85487,87249,90749
%N A192297 Lesser of pseudo twin primes to base 2.
%C A192297 We call numbers {k,k+2} pseudo twin primes to base 2 if at least one of them is composite, while 2^(k-1) == 1 (mod k) and 2^(k+1) == 1 mod (k+2).
%C A192297 4369 is the only known term such that both k and k+2 are composite (cf. A173619). - _Jianing Song_, Nov 20 2021
%H A192297 Amiram Eldar, <a href="/A192297/b192297.txt">Table of n, a(n) for n = 1..15587</a> (terms below 10^12; terms 1..1000 from Alois P. Heinz)
%F A192297 2^(a(n) + 2) == 3*a(n) + 8 (mod a(n)*(a(n)+2)).
%F A192297 4*(2^(a(n)-1)-1) == -a(n)*((a(n)-1)/2) (mod a(n)*(a(n)+2)). - _Davide Rotondo_, Nov 07 2021
%p A192297 a:= proc(n) option remember; local k;
%p A192297       for k from 2+`if` (n=1, 1, a(n-1)) by 2 while
%p A192297         isprime(k) and isprime(k+2) or
%p A192297           (2&^(k-1) mod k)<>1 or (2&^(k+1) mod (k+2))<>1
%p A192297       do od; k
%p A192297     end:
%p A192297 seq (a(n), n=1..40);  # _Alois P. Heinz_, Oct 13 2011
%t A192297 fQ[n_] := (! PrimeQ[n] || ! PrimeQ[n + 2]) && PowerMod[2, n - 1, n] == 1 && PowerMod[2, n + 1, n + 2] == 1; Select[2 Range@ 32000 + 1, fQ] (* _Robert G. Wilson v_, Oct 11 2011 *)
%o A192297 (PARI) is(n)=Mod(2,n^2+2*n)^(n+2)==3*n+8 && (!isprime(n) || !isprime(n+2)) && n>1 \\ _Charles R Greathouse IV_, Dec 02 2014
%Y A192297 Cf. A001567, A002997, A141232.
%Y A192297 Cf. A173619.
%K A192297 nonn
%O A192297 1,1
%A A192297 _Vladimir Shevelev_, Oct 11 2011