A163209 - OEIS

%I #35 May 08 2020 17:32:24

%S 5907,1194649,12327121

%N Catalan pseudoprimes: odd composite integers n=2*m+1 satisfying A000108(m) == (-1)^m * 2 (mod n).

%C 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.

%C Also, composite odd integers n=2*m+1 such that A000984(m) == (-1)^m (mod n).

%C 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).

%C See the Vardi reference for a binomial setup.

%C Aebi and Cairns 2008, page 9: a(4) either has more than 2 factors or is > 10^10. - _Dana Jacobsen_, May 27 2015

%C a(4) > 10^10. - _Dana Jacobsen_, Mar 03 2018

%D I. Vardi, Computational Recreations in Mathematica, 1991, p. 66.

%H C. Aebi, G. Cairns (2008). "<a href="http://gradelle.educanet2.ch/christian.aebi/.ws_gen/9/catalan.pdf">Catalan numbers, primes and twin primes</a>". Elemente der Mathematik 63 (4): 153-164. doi:<a href="http://dx.doi.org/10.4171/EM/103">10.4171/EM/103</a>

%H Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.

%H Peter Luschny, <a href="http://www.luschny.de/math/primes/SwingingPrimes.html"> Swinging Primes.</a>

%H <a href="/index/Ps#pseudoprimes">Index entries for sequences related to pseudoprimes</a>

%p 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:

%p WS := proc(f,r,n) select(p->(f(p-1)+r(p)) mod p = 0,[$2..n]);

%p select(q -> not isprime(q),%) end:

%p A163209 := n -> WS(swing,p->(-1)^iquo(p+2,2),n);

%o (PARI) v(n,p)=my(s); n*=2; while(n\=p, s+=n%2); s

%o 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

%o forcomposite(n=4,2e7, if(is(n), print1(n", "))) \\ _Charles R Greathouse IV_, Mar 06 2015

%o (Perl) # Reasonable for isolated values, slow for the sequence:

%o use ntheory ":all";

%o sub is { my $m = ($_[0]-1)>>1; (binomial($m<<1,$m) % $_[0]) == (($m&1) ? $_[0]-1 : 1); }

%o foroddcomposites { say if is($_) } 2e7;  # _Dana Jacobsen_, May 03 2015

%o (Perl) # Much faster for sequential testing:

%o 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

%K nonn,hard,more,bref

%O 1,1

%A _Peter Luschny_, Jul 24 2009

%E a(1) = 5907 = 3*11*179 was found by S. Skiena

%E Typo corrected _Peter Luschny_, Jul 25 2009

%E Edited by _Max Alekseyev_, Jun 22 2012