A290059 a(n) = binomial(2*prime(n)-1, prime(n)-1) where prime(n) is the n-th prime.

3, 10, 126, 1716, 352716, 5200300, 1166803110, 17672631900, 4116715363800, 15033633249770520, 232714176627630544, 873065282167813104916, 212392290424395860814420, 3318776542511877736535400, 812850570172585125274307760, 3136262529306125724764953838760

Offset: 1

Author

Martin Renner, Jul 19 2017

Keywords

• nonn

Comments

Charles Babbage (1791-1871) proved in 1819 that for every prime p > 2 this sequence is congruent to 1 (mod p^2).

Joseph Wolstenholme (1829-1891) proved in 1862 that for every prime p > 3 this sequence is congruent to 1 (mod p^3).

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..261
• Charles Babbage, Demonstration of a theorem relating to prime numbers, The Edinburgh philosophical journal 1 (1819), p. 46-49.
• Joseph Wolstenholme, On certain properties of prime numbers, The quarterly journal of pure and applied mathematics 5 (1862), p. 35-39.

Crossrefs

Cf. A088164. Subsequence of A001700.

Programs

Maple

seq(binomial(2*ithprime(i)-1,ithprime(i)-1),i=1..16);

Mathematica

Array[Function[p, Binomial[2*p - 1, p - 1]]@ Prime@ # &, 16] (* Michael De Vlieger, Jul 19 2017 *)
Binomial[2#-1,#-1]&/@Prime[Range[20]] (* Harvey P. Dale, Mar 29 2024 *)

PARI

a(n) = my(p=prime(n)); binomial(2*p-1, p-1); \\ Michel Marcus, Jul 19 2017

Python

from sympy import prime, binomial
def a(n):
    p=prime(n)
    return binomial(2*p - 1, p - 1)
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jul 20 2017

Formula

log(a(n)) ~ 2*log(2)*n * (log(n) + log(log(n)) - 1). - Vaclav Kotesovec, May 07 2022