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 A163209 Catalan pseudoprimes: odd composite integers n=2*m+1 satisfying A000108(m) == (-1)^m * 2 (mod n). 6
 5907, 1194649, 12327121 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Also, Wilson spoilers: composite n which divide A056040(n-1) - (-1)^floor(n/2). For the factorial function, a Wilson spoiler is a composite n that divides (n-1)! + (-1). Lagrange proved that no such n exists. For the swinging factorial (A056040), the situation is different. Also, composite odd integers n=2*m+1 such that A000984(m) == (-1)^m (mod n). Contains squares of A001220. In particular, a(2) = A001220(1)^2 = 1093^2 = 1194649 = A001567(274) and a(3) = A001220(2)^2 = 3511^2 = 12327121 = A001567(824). See the Vardi reference for a binomial setup. Aebi and Cairns 2008, page 9: a(4) either has more than 2 factors or is > 10^10. - Dana Jacobsen, May 27 2015 a(4) > 2*10^9. - Dana Jacobsen, Jun 01 2015 REFERENCES Peter Luschny, "Divide, swing and conquer the factorial and the lcm{1,2,...,n}", preprint, April 2008. I. Vardi, Computational Recreations in Mathematica, 1991, p. 66. LINKS C. Aebi, G. Cairns (2008). "Catalan numbers, primes and twin primes". Elemente der Mathematik 63 (4): 153-164. doi:10.4171/EM/103 Peter Luschny, Swinging Primes. MAPLE swing := proc(n) option remember; if n = 0 then 1 elif irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end: WS := proc(f, r, n) select(p->(f(p-1)+r(p)) mod p = 0, [\$2..n]); select(q -> not isprime(q), %) end: A163209 := n -> WS(swing, p->(-1)^iquo(p+2, 2), n); PROG (PARI) v(n, p)=my(s); n*=2; while(n\=p, s+=n%2); s is(n)=if(n%2==0, return(0)); my(m=Mod(1, n), a=n\2); fordiv(n, d, if(isprime(d) && v(a, d), return(0))); forprime(p=2, a, m*=p^v(a, p)); forprime(p=a+1, n, m*=p); m==(-1)^a forcomposite(n=4, 2e7, if(is(n), print1(n", "))) \\ Charles R Greathouse IV, Mar 06 2015 (Perl) # Reasonable for isolated values, slow for the sequence: use ntheory ":all"; sub is { my \$m = (\$_[0]-1)>>1; (binomial(\$m<<1, \$m) % \$_[0]) == ((\$m&1) ? \$_[0]-1 : 1); } foroddcomposites { say if is(\$_) } 2e7;  # Dana Jacobsen, May 03 2015 (Perl) # Much faster for sequential testing: use Math::GMPz; use ntheory ":all"; { my(\$c, \$l)=(Math::GMPz->new(1), 1); sub catalan { while (\$_[0] > \$l) { \$l++; \$c *= 4*\$l-2; Math::GMPz::Rmpz_divexact_ui(\$c, \$c, \$l+1); } \$c; } } my \$m; foroddcomposites { \$m = (\$_-1)>>1; say if (catalan(\$m) % \$_) == ((\$m&1) ? \$_-2 : 2); } 2e7;  # Dana Jacobsen, May 03 2015 CROSSREFS Sequence in context: A025513 A015295 A209431 * A216942 A252289 A251464 Adjacent sequences:  A163206 A163207 A163208 * A163210 A163211 A163212 KEYWORD nonn,hard,more,bref AUTHOR Peter Luschny, Jul 24 2009 EXTENSIONS a(1) = 5907 = 3*11*179 was found by S. Skiena Typo corrected Peter Luschny, Jul 25 2009 Edited by Max Alekseyev, Jun 22 2012 STATUS approved

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