A086001 - OEIS

%I #11 Mar 27 2021 08:00:07

%S 93,32,41,3,7,4,4,4,3,1,2,8,1,2,3,4,8,2,4,1,5,13,1,4,6,3,60,1,1,1,21,

%T 8,74,4,1,1,16,3,7,793,7,12,1,17,7,9,24,15,5,1,85,4,1,1,4,2155,3,1,1,

%U 25,6,1,27,1,1669,1,1,12,6,1,4,57,15,29,817,4,2,3,4,63,3,20,1,12,3,11,3,9,31

%N For p = prime(n), a(n) is the smallest k such that p*(1 + 2 Ord(4,p) k) is a base-2 pseudoprime.

%C Sequences A085999 and A086000 list p*(1 + 2 Ord(4,p) k) and 1 + 2 Ord(4,p) k., respectively. Although, for any prime p, Dirichlet's theorem says the sequence 1 + 2 Ord(4,p) k contains an infinite number of primes, only a finite number of these produce a pseudoprime when multiplied by p.

%H Amiram Eldar, <a href="/A086001/b086001.txt">Table of n, a(n) for n = 2..10001</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Pseudoprime.html">Pseudoprime</a>.

%e a(11) = 1 because prime(11) = 31, ord(4,31) = 5 and 31*(1+2*5*1) is a 2-pseudoprime.

%t Table[p=Prime[n]; m=MultiplicativeOrder[4, p]; k=1; While[psp=p(1+2*m*k); PowerMod[2, psp-1, psp]!=1, k++ ]; k, {n, 2, 100}]

%Y Cf. A001567 (base-2 pseudoprimes), A082654 (ord(4, p)), A085012, A085999, A086000.

%K easy,nonn

%O 2,1

%A _T. D. Noe_, Jul 08 2003