A109760 Composite numbers k such that binomial(5*k, k) == 5^k (mod k).

4, 365, 400, 685, 3200, 6400, 12550, 12800, 16525, 25600, 51200, 225125, 70463125, 271094125, 431434441, 1639197169, 2434638125, 3050358125

Offset

1,1

Comments

72154484375 is a term. - Chai Wah Wu, Jan 13 2026

p^2 is a term if p is in A123692, i.e. 2859165058687921, 2707122349256919049, 44787780573344471569, 35625862920811846533601 are terms. - Chai Wah Wu, Jan 14 2026

7924899803125 is a term. - Chai Wah Wu, Jan 31 2026

Links

Table of n, a(n) for n=1..18.

Max Alekseyev, PARI/GP scripts for various problems

Example

4 is a term because binomial(5*4, 4) = 4845, 5^4 = 625 and 4845 mod 4 = 625 mod 4 = 1.

Mathematica

Do[If[ !PrimeQ[n], If[Mod[Binomial[5*n, n], n] == Mod[5^n, n], Print[n]]], {n, 1, 50000}]
Select[Range[250000], CompositeQ[#]&&Mod[Binomial[5#, #], #]==PowerMod[5, #, #]&] (* The program generates the first 12 terms of the sequence. *) (* Harvey P. Dale, Jul 28 2025 *)

Prog

(Python)
from itertools import count, islice
from sympy import isprime
from oeis_sequences.OEISsequences import binom_mod
def A109760_gen(startvalue=4): # generator of terms >= startvalue
    for j in count(max(startvalue, 4)):
        if not isprime(j) and binom_mod(5*j, j, j) == pow(5, j, j):
            yield j
A109760_list = list(islice(A109760_gen(), 6)) # Chai Wah Wu, Jan 13 2026

Crossrefs

Cf. A080469, A123692.

Sequence in context: A366469 A051955 A177114 * A051181 A326996 A385885

Adjacent sequences: A109757 A109758 A109759 * A109761 A109762 A109763

Keyword

hard,more,nonn

Author

Ryan Propper, Aug 12 2005

Extensions

a(11) from D. S. McNeil, Mar 15 2009

a(12) from Max Alekseyev, Sep 13 2009

a(13)-a(15) from Max Alekseyev, Nov 06 2009

a(16) from Chai Wah Wu, Jan 14 2026

a(17)-a(18) from Chai Wah Wu, Jan 15 2026

Status

approved