A086249 Number of base-2 Fermat pseudoprimes x that have ord(2,x) = n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 3, 1, 2, 1, 1, 0, 12, 4, 3, 0, 1, 1, 1, 1, 12, 1, 1, 4, 5, 1, 9, 4, 10, 8, 3, 4, 25, 0, 10, 11, 11, 4, 1, 4, 15, 4, 22, 1, 57, 0, 1, 4, 10, 1, 24, 1, 11, 1, 41, 4, 86, 4, 10, 25, 11, 0, 21, 4, 7, 4, 10, 1, 52, 1, 7, 10, 22, 0, 26, 11, 56, 1

Offset: 1

Author

T. D. Noe, Jul 14 2003

Keywords

• hard
• nonn

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 rA086250 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.

Examples

			a(10) = 1 there is only 1 pseudoprime, 341 = 11*31, having order 10; that is, 2^10 = 1 mod 341.
		

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

Crossrefs

Cf. A001567 (base-2 pseudoprimes), A086250.

Programs

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}]

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 */