OFFSET
1,22
COMMENTS
A base-2 Fermat pseudoprime is a composite number x such that 2^x == 2 (mod x). For such an x, ord(2,x) is the smallest positive integer m such that 2^m == 1 (mod x). For a number x to have order n, it must be a factor of 2^n-1 and not be a factor of 2^r-1 for r<n. Sequence A086250 lists the smallest pseudoprime of order n.
Note that there is no pseudoprime of order n when 2^n-1 is prime. However that does not explain why there are none for 12, 27, 49 and 77.
LINKS
Max Alekseyev, Table of n, a(n) for n = 1..200
R. G. E. Pinch, Pseudoprimes and their factors (FTP)
Eric Weisstein's World of Mathematics, Pseudoprime
EXAMPLE
a(10) = 1 there is only 1 pseudoprime, 341 = 11*31, having order 10; that is, 2^10 = 1 mod 341.
MATHEMATICA
Table[d=Divisors[2^n-1]; cnt=0; Do[m=d[[i]]; If[ !PrimeQ[m]&&PowerMod[2, m, m]==2&&MultiplicativeOrder[2, m]==n, cnt++ ], {i, Length[d]}]; cnt, {n, 100}]
PROG
(PARI) { a(n) = my(r=0); fordiv(2^n-1, d, if(d>1 && (d-1)%n==0 && !ispseudoprime(d) && znorder(Mod(2, d), n)==n, r++) ); r } /* Max Alekseyev, Jan 07 2015 */
CROSSREFS
KEYWORD
hard,nonn
AUTHOR
T. D. Noe, Jul 14 2003
STATUS
approved