A227905 Numbers of the form 4k+3 (A004767) that are Lucas pseudoprimes and Fermat pseudoprimes to base 2 (intersection of A005845 and A001567).

741751, 1024651, 5481451, 31150351, 109437751, 139952671, 178482151, 284301751, 383425351, 395044651, 407282851, 417027451, 498706651, 582799951, 612816751, 620072251, 652969351, 738820351, 977755351, 1126587151, 1204176751, 1397357851, 1588247851, 1789167931

Offset: 1

Author

José María Grau Ribas, Oct 12 2013

Keywords

• nonn

Comments

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• J. M. Grau, A. M. Oller-Marcen, M. Rodríguez, D. Sadornil, Fermat test with gaussian base and Gaussian pseudoprimes, arXiv preprint arXiv:1401.4708 [math.NT], 2014.
• W. Galway, Tables of pseudoprimes and related data [Includes a file with base-2 Fermat pseudoprimes up to 2^64.]

Crossrefs

Cf. A004767 (4n+3).

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

Programs

Mathematica

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

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

Perl

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

Extensions

More terms from Dana Jacobsen, Jan 07 2015

a(16)-a(24) from Amiram Eldar, Jun 27 2019