A343533 a(n) is the largest value of k such that binomial(2*m-1, m-1) == 1 (mod m^k) for m = 2*n + 1.

2, 3, 3, 1, 3, 3, 0, 3, 3, 0, 3, 1, 0, 3, 3, 0, 0, 3, 0, 3, 3, 0, 3, 1, 0, 3, 0, 0, 3, 3, 0, 0, 3, 0, 3, 3, 0, 0, 3, 0, 3, 0, 0, 3, 0, 0, 0, 3, 0, 3, 3, 0, 3, 3, 0, 3, 0, 0, 0, 1, 0, 1, 3, 0, 3, 0, 0, 3, 3, 0, 0, 0, 0, 3, 3, 0, 0, 3, 0, 0, 3, 0, 3, 1, 0, 3, 0

Offset: 1

Author

Felix Fröhlich, Apr 18 2021

Keywords

• nonn

Comments

If 2*n + 1 is a prime >= 5, then a(n) >= 3 by Wolstenholme's theorem.

If 2*n + 1 is a Wolstenholme prime (A088164), then a(n) >= 4.

If 2*n + 1 is a term of A267824, then a(n) >= 2.

If 2*n + 1 is the square of an odd prime, the cube of a prime >= 5 or a term of A228562, then a(n) >= 1.

Crossrefs

Cf. A005408, A088164, A136327, A228562, A267824.

Programs

Maple

a := proc(n) local x, x0, y, k, bound; bound := 1000;
x := 2*n + 1; x0 := x;
y := binomial(4*n + 1, 2*n);
for k from 0 to bound while y mod x = 1 do
    x := x * x0 od;
if k < bound then k else print("No k below ", bound) fi end:
seq(a(n), n = 1..100); # Peter Luschny, Apr 22 2021

PARI

a(n) = my(x=2*n+1, b=binomial(2*x-1, x-1)); for(k=1, oo, if(Mod(b, x^k)!=1, return(k-1)))