cp's OEIS Frontend

A182154 Smallest k >= 2 such that k^(2^n)+1 is the lesser member of a twin prime pair.

%I A182154 #21 Jun 12 2025 03:39:47
%S A182154 2,2,2,4,2,49592,7132,532,333482,2226686,3543554,23379038,1249625230,
%T A182154 188489906
%N A182154 Smallest k >= 2 such that k^(2^n)+1 is the lesser member of a twin prime pair.
%C A182154 These lesser of twin prime pairs are also generalized Fermat primes, (not possible for greater of twin prime pairs, except for 5).
%C A182154 When extending this sequence, it is useful if the primes b^(2^n)+1 are known in advance (Gallot link). - _Jeppe Stig Nielsen_, Sep 25 2019
%C A182154 For later terms, the bigger twin is only a probable prime, not a proven prime. - _Jeppe Stig Nielsen_, Nov 24 2022
%H A182154 Yves Gallot, <a href="http://yves.gallot.pagesperso-orange.fr/primes/results.html">Generalized Fermat Prime Search</a>.
%H A182154 OEIS Wiki, <a href="/wiki/Generalized_Fermat_numbers">Generalized Fermat numbers</a>.
%H A182154 PrimeGrid and "Stream", <a href="https://www.primegrid.com/forum_thread.php?id=9538">GFN-1x Small Primes search</a>, mentions a(12) and a(13).
%e A182154 2^(2^4)+1 = 65537 = A001359(861), then a(4) = 2.
%t A182154 Table[k=2; While[!PrimeQ[k^(2^n)+1]||!PrimeQ[k^(2^n)+3],k++]; k,{n,0,7}]
%Y A182154 Cf. A056993, A001359.
%K A182154 nonn,more,hard
%O A182154 0,1
%A A182154 _Manuel Valdivia_, Apr 15 2012
%E A182154 a(8)-a(10) from _Jeppe Stig Nielsen_, Sep 25 2019
%E A182154 Name edited by _Felix Fröhlich_, Sep 25 2019
%E A182154 a(11)-a(13) from _Jeppe Stig Nielsen_, Nov 24 2022