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%I #6 Jan 07 2015 10:38:37
%S 0,0,0,0,0,0,0,0,0,341,2047,0,0,5461,4681,4369,0,1387,0,13981,42799,
%T 15709,8388607,1105,1082401,22369621,0,645,256999,10261,0,16843009,
%U 1227133513,5726623061,8727391,1729,137438953471,91625968981,647089,561
%N Smallest base-2 Fermat pseudoprime x that has ord(2,x) = n, or 0 if one does not exist.
%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 A086249 lists the number of pseudoprimes of order n.
%H Max Alekseyev, <a href="/A086250/b086250.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]; num=0; i=1; done=False; While[m=d[[i]]; done=!PrimeQ[m]&&PowerMod[2, m, m]==2&&MultiplicativeOrder[2, m]==n; If[done, num=m]; !done&&i<Length[d], i++ ]; num, {n, 100}]
%o (PARI) { a(n) = fordiv(2^n-1,d, if(d>1 && (d-1)%n==0 && !ispseudoprime(d) && znorder(Mod(2,d))==n,return(d)) ); 0 } /* _Max Alekseyev_, Jan 07 2015 */
%Y Cf. A001567 (base-2 pseudoprimes), A086249.
%K hard,nonn
%O 1,10
%A _T. D. Noe_, Jul 14 2003
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