A080469 Composite numbers k such that binomial(3*k, k) == 3^k (mod k).

36, 57, 121, 132, 552, 8397, 7000713, 9692541, 36294723, 564033861, 11525933961

Offset

1,1

Comments

If p is prime, binomial(3*p,p) == 3^p (mod p).

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

285741430137 and 25788824595441 are terms. p^2 is a term if p is in A014127, i.e. 1012042036009 is a term. - Chai Wah Wu, Jan 14 2026

Links

Table of n, a(n) for n=1..11.

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
(Python)
from itertools import count, islice
from sympy import isprime
from oeis_sequences.OEISsequences import binom_mod
def A080469_gen(startvalue=4): # generator of terms >= startvalue
    for j in count(max(startvalue, 4)):
        if not isprime(j) and binom_mod(3*j, j, j) == pow(3, j, j):
            yield j
A080469_list = list(islice(A080469_gen(), 6)) # Chai Wah Wu, Jan 13 2026

Crossrefs

Cf. A014127, A109641, A109642; A109760, A109769.

Sequence in context: A124941 A116321 A187989 * A341283 A260138 A260131

Adjacent sequences: A080466 A080467 A080468 * A080470 A080471 A080472

Keyword

nonn,more,hard

Author

Benoit Cloitre, Oct 15 2003

Extensions

a(6) from Ryan Propper, Aug 12 2005

a(7)-a(10) from Max Alekseyev, Nov 05 2009

a(11) from Chai Wah Wu, Jan 26 2026

Status

approved