

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



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



KEYWORD

nonn,more,hard


AUTHOR



EXTENSIONS



STATUS

approved



