A263882 Babbage quotients b_p = (binomial(2p-1, p-1) - 1)/p^2 with p = prime(n).

1, 5, 35, 2915, 30771, 4037381, 48954659, 7782070631, 17875901604959, 242158352370063, 637739431824553035, 126348774791431208099, 1794903484322270273951, 367972191114796344623951, 1116504994413003106003899551, 3498520498083111051973370669639

Offset: 2

Author

Jonathan Sondow, Nov 22 2015

Keywords

• nonn

Comments

Charles Babbage proved in 1819 that b_p is an integer for prime p > 2. In 1862 Wolstenholme proved that the Wolstenholme quotient W_p = b_p / p is an integer for prime p > 3; see A034602.

The quotient b_n is an integer for composite n in A267824. No composite n is known for which W_n is an integer.

Examples

			a(2) = (binomial(2*3-1,3-1) - 1)/3^2 = (binomial(5,2) - 1)/9 = (10-1)/9 = 1.
		

References

• R. K. Guy, Unsolved Problems in Number Theory, Sect. B31.

Links

• Robert Israel, Table of n, a(n) for n = 2..260
• C. Babbage, Demonstration of a theorem relating to prime numbers, Edinburgh Philosophical Journal, 1 (1819), 46-49.
• R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
• J. Sondow, Extending Babbage's (non-)primality tests, in Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, 269-277, CANT 2015 and 2016, New York, 2017; arXiv:1812.07650 [math.NT], 2018.
• Wikipedia, Wolstenholme's theorem
• J. Wolstenholme, On certain properties of prime numbers, Quarterly Journal of Pure and Applied Mathematics, 5 (1862), 35-39.

Crossrefs

Cf. A034602, A088164, A099906, A267824.

Programs

Magma

[(Binomial(2*NthPrime(n)-1, NthPrime(n)-1)-1)/NthPrime(n)^2: n in [2..20]]; // Vincenzo Librandi, Nov 25 2015

Maple

map(p -> (binomial(2*p-1,p-1)-1)/p^2, select(isprime,[seq(i,i=3..100,2)])); # Robert Israel, Nov 24 2015

Mathematica

Table[(Binomial[2*Prime[n] - 1, Prime[n] - 1] - 1)/Prime[n]^2, {n, 2, 17}]
Table[(Binomial[2p-1,p-1]-1)/p^2,{p,Prime[Range[2,20]]}] (* Harvey P. Dale, Jul 20 2019 *)

Formula

a(n) = prime(n)*A034602(n) for n > 2.

a(PrimePi(A088164(n))) == 0 mod A088164(n)^2.