

A080469


Composite n such that binomial(3*n,n)==3^n (mod n).


7




OFFSET

1,1


COMMENTS

If p is prime, binomial(3*p,p)==3^p (mod p)
No other terms below 10^9.
A subsequence of A109641. The terms a(n) with n=2, 6, 7, 8, 9, 10 are of the form 3^k*p where p is prime and k=1, 3, 2, 5, 6, 7, respectively. It is tempting to conjecture that there are (infinitely many?) more terms of that form.  M. F. Hasler, Nov 11 2015


LINKS

Table of n, a(n) for n=1..10.
Max Alekseyev, PARI scripts for various problems


EXAMPLE

57 is a term because binomial(3*57, 57) = 12039059761216294940321619222324879408784636200 mod 57 = 27 == 3^57 mod 57.


MATHEMATICA

Do[If[ !PrimeQ[n], k = Binomial[3*n, n]; m = 3^n; If[Mod[k, n] == Mod[m, n], Print[n]]], {n, 1, 70000}] (* Ryan Propper, Aug 12 2005 *)


PROG

(PARI) forcomposite(n=1, 1e9, binomod(3*n, n, n)==Mod(3, n)^n && print1(n", ")) \\ Cf. Alekseyev link.  M. F. Hasler, Nov 14 2015


CROSSREFS

Cf. A109641, A109642; A109760, A109769.
Sequence in context: A124941 A116321 A187989 * A260138 A260131 A188633
Adjacent sequences: A080466 A080467 A080468 * A080470 A080471 A080472


KEYWORD

nonn,more,hard


AUTHOR

Benoit Cloitre, Oct 15 2003


EXTENSIONS

One more term a(6) from Ryan Propper, Aug 12 2005
Four new terms a(7)a(10) added by Max Alekseyev, Nov 05 2009


STATUS

approved



