cp's OEIS Frontend

A086019 For p = prime(n), a(n) is the largest prime q such that pq is a base-2 pseudoprime; that is, 2^(pq-1) = 1 mod pq; a(n) is 0 if no such prime exists.

%I A086019 #14 Mar 27 2021 08:00:04
%S A086019 0,0,0,31,0,257,73,683,113,331,109,61681,5419,2796203,1613,3033169,
%T A086019 1321,599479,122921,38737,22366891,8831418697,2931542417,22253377,
%U A086019 268501,131071,28059810762433,279073,54410972897,77158673929,145295143558111
%N A086019 For p = prime(n), a(n) is the largest prime q such that pq is a base-2 pseudoprime; that is, 2^(pq-1) = 1 mod pq; a(n) is 0 if no such prime exists.
%C A086019 Using a theorem of Lehmer, it can be shown that the possible values of q are among the prime factors of 2^(p-1)-1. Sequence A085012 gives the smallest prime q such that 2^(pq-1) = 1 mod pq. Sequence A085014 gives the number of 2-factor pseudoprimes that have prime(n) as a factor.
%D A086019 Paulo Ribenboim, The New Book of Prime Number Records, Springer, 1996, p. 105-112.
%H A086019 Amiram Eldar, <a href="/A086019/b086019.txt">Table of n, a(n) for n = 2..337</a>
%H A086019 Paul Erdős, <a href="http://www.jstor.org/stable/2304732">On the converse of Fermat's theorem</a>, Amer. Math. Monthly 56 (1949), p. 623-624.
%H A086019 D. H. Lehmer, <a href="http://www.jstor.org/stable/2301798">On the converse of Fermat's theorem</a>, Amer. Math. Monthly 43 (1936), p. 347-354.
%e A086019 a(9) = 683 because prime(9) = 23 and 683 is the largest factor of 2^22-1 that yields a pseudoprime when multiplied by 23.
%t A086019 Table[p=Prime[n]; q=Reverse[Transpose[FactorInteger[2^(p-1)-1]][[1]]]; i=1; While[i<=Length[q]&&(PowerMod[2, p*q[[i]]-1, p*q[[i]]]>1), i++ ]; If[i>Length[q], 0, q[[i]]], {n, 2, 40}]
%Y A086019 Cf. A001567 (base 2 pseudoprimes), A085012, A085014, A180471.
%K A086019 hard,nonn
%O A086019 2,4
%A A086019 _T. D. Noe_, Jul 08 2003