

A163209


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


5




OFFSET

1,1


COMMENTS

Also, Wilson spoilers: composite n which divide A056040(n1)  (1)^floor(n/2). For the factorial function, a Wilson spoiler is a composite n that divides (n1)! + (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.


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

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): 153164. 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(n1)*n else 4*swing(n1)/n fi end:
WS := proc(f, r, n) select(p>(f(p1)+r(p)) mod p = 0, [$2..n]);
select(q > not isprime(q), %) end:
A163209 := n > WS(swing, p>(1)^iquo(p+2, 2), n);


CROSSREFS

Sequence in context: A025513 A015295 A209431 * A216942 A031665 A031575
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



