A109769 Composite numbers k such that binomial(7*k, k) == 7^k (mod k).

18, 25, 133, 2107, 4676, 226037, 4477739, 827867201, 39103456721

Offset

1,1

Comments

106164747059 is a term. p^2 is a term if p is in A123693, i.e. 241602723961 is a term. - Chai Wah Wu, Jan 14 2026

46303016911543 is a term. - Chai Wah Wu, Feb 11 2026

Links

Table of n, a(n) for n=1..9.

Max Alekseyev, PARI/GP scripts for various problems

Example

18 is a term because binomial(7*18, 18) = 2797093093529137508875, 7^18 = 1628413597910449 and 2797093093529137508875 mod 18 = 1628413597910449 mod 18 = 1.

Mathematica

Do[If[ !PrimeQ[n], If[Mod[Binomial[7*n, n], n] == Mod[7^n, n], Print[n]]], {n, 2, 20000}]

Prog

(Python)
from itertools import count, islice
from sympy import isprime
from oeis_sequences.OEISsequences import binom_mod
def A109769_gen(startvalue=4): # generator of terms >= startvalue
    for j in count(max(startvalue, 4)):
        if not isprime(j) and binom_mod(7*j, j, j) == pow(7, j, j):
            yield j
A109769_list = list(islice(A109769_gen(), 6)) # Chai Wah Wu, Jan 13 2026
(PARI) isok(k) = if (!isprime(k) && (k>1), binomod(7*k, k, k) == Mod(7, k)^k); \\ Michel Marcus, Feb 13 2026, using Max Alekseyev's binomod

Crossrefs

Cf. A080469, A123693.

Sequence in context: A393026 A362435 A248111 * A350773 A229967 A212049

Adjacent sequences: A109766 A109767 A109768 * A109770 A109771 A109772

Keyword

nonn,hard,more

Author

Ryan Propper, Aug 13 2005

Extensions

a(6) from Max Alekseyev, Sep 13 2009

a(7)-a(8) from Max Alekseyev, Nov 06 2009

a(9) from Chai Wah Wu confirmed and added by Max Alekseyev, Feb 13 2026

Status

approved