A390043 Composite numbers k such that binomial(4*k, k) == 4^k (mod k).

4, 6, 18, 30, 39, 54, 56, 68, 84, 92, 184, 208, 992, 2704, 16256, 27424, 52064, 129848, 225229, 514688, 616688, 1194649, 1818176, 3788416, 12327121, 53703248, 67100672, 2160093568, 2779417328, 3899921104

Offset

1,1

Comments

If k is prime, then binomial(4*k, k) == 4^k (mod k).

If p is a prime such that p^2 divides 4^(p-1) - 1, then p^2 is a term.

Links

Table of n, a(n) for n=1..30.

Prog

(Python)
from itertools import count, islice
from sympy import isprime
from oeis_sequences.OEISsequences import binom_mod
def A390043_gen(startvalue=4): # generator of terms >= startvalue
    for j in count(max(startvalue, 4)):
        if not isprime(j) and binom_mod(j<<2, j, j) == pow(4, j, j):
            yield j
A390043_list = list(islice(A390043_gen(), 20))
(PARI) isok(k) = !isprime(k) && binomod(4*k, k, k)==Mod(4, k)^k; \\ Michel Marcus, Jan 16 2026; using binomod.gp by M. Alekseyev

Crossrefs

Cf. A002808, A084699, A080469, A109760, A109769, A392484.

Sequence in context: A064217 A026623 A026689 * A138276 A287682 A209236

Adjacent sequences: A390040 A390041 A390042 * A390044 A390045 A390046

Keyword

nonn,hard,more

Author

Chai Wah Wu, Jan 15 2026

Extensions

a(29)-a(30) from Chai Wah Wu, Jan 19 2026

Status

approved