A227905 - OEIS

%I #40 Jun 28 2019 14:42:43

%S 741751,1024651,5481451,31150351,109437751,139952671,178482151,

%T 284301751,383425351,395044651,407282851,417027451,498706651,

%U 582799951,612816751,620072251,652969351,738820351,977755351,1126587151,1204176751,1397357851,1588247851,1789167931

%N Numbers of the form 4k+3 (A004767) that are Lucas pseudoprimes and Fermat pseudoprimes to base 2 (intersection of A005845 and A001567).

%C This sequence uses the Bruckman definition of "Lucas pseudoprime". There are 400,114 examples less than 2^64. - _Dana Jacobsen_, Jan 07 2015

%H Amiram Eldar, <a href="/A227905/b227905.txt">Table of n, a(n) for n = 1..10000</a>

%H J. M. Grau, A. M. Oller-Marcen, M. Rodríguez, D. Sadornil, <a href="http://arxiv.org/abs/1401.4708">Fermat test with gaussian base and Gaussian pseudoprimes</a>, arXiv preprint arXiv:1401.4708 [math.NT], 2014.

%H W. Galway, <a href="http://www.cecm.sfu.ca/Pseudoprimes/index-2-to-64.html">Tables of pseudoprimes and related data</a> [Includes a file with base-2 Fermat pseudoprimes up to 2^64.]

%t Select[4*Range[8000000]+3, CompositeQ[#] && PowerMod[2, (# - 1), # ] == 1 && Divisible[LucasL[#]-1, #] &] (* _Amiram Eldar_, Jun 27 2019 *)

%o # Using the Feitsma/Galway database from the links:

%o (Perl) perl -Mntheory=:all -nE 'chomp; say if ($_%4)==3 && (lucas_sequence($_,1,-1,$_))[1] == 1' psps-below-2-to-64.txt  # _Dana Jacobsen_, Jan 07 2015

%o (Perl) perl -Mntheory=:all -E 'foroddcomposites { say if $_%4 == 3 && ispseudoprime($_,2) && (lucas_sequence($_,1,-1,$_))[1] == 1 } 1e14'  # _Dana Jacobsen_, Jan 10 2015

%Y Cf. A004767 (4n+3).

%Y Cf. A001567 (Fermat pseudoprimes to base 2), A005845 (Lucas pseudoprimes).

%K nonn

%O 1,1

%A _José María Grau Ribas_, Oct 12 2013

%E More terms from _Dana Jacobsen_, Jan 07 2015

%E a(16)-a(24) from _Amiram Eldar_, Jun 27 2019