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) > 10^10. - Dana Jacobsen, Mar 03 2018
REFERENCES
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, Die schwingende Fakultät und Orbitalsysteme, August 2011.
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
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