A086249 - OEIS

%I #11 Feb 11 2015 06:43:53

%S 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,

%T 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,

%U 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

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

%C 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.

%C 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.

%H Max Alekseyev, <a href="/A086249/b086249.txt">Table of n, a(n) for n = 1..200</a>

%H R. G. E. Pinch, <a href="ftp://ftp.dpmms.cam.ac.uk/pub/PSP/">Pseudoprimes and their factors (FTP)</a>

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

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

%t 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}]

%o (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 */

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

%K hard,nonn

%O 1,22

%A _T. D. Noe_, Jul 14 2003