A163209 Catalan pseudoprimes: odd composite integers n=2*m+1 satisfying A000108(m) == (-1)^m * 2 (mod n).

5907, 1194649, 12327121

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

Table of n, a(n) for n=1..3.

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.

Index entries for sequences related to pseudoprimes

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 A284980

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