A306204 Decimal expansion of Product_{p>=3} (1+1/p) over the Mersenne primes.

1, 5, 8, 5, 5, 5, 8, 8, 8, 7, 9, 2, 5, 6, 3, 8, 7, 7, 6, 9, 7, 8, 6, 3, 7, 0, 2, 3, 2, 1, 9, 2, 3, 8, 4, 7, 6, 0, 6, 9, 4, 0, 5, 8, 6, 7, 9, 4, 7, 0, 2, 8, 1, 1, 3, 2, 9, 8, 1, 2, 6, 7, 8, 9, 2, 8, 8, 5, 9, 7, 5, 4, 5, 7, 6, 7, 8, 5, 5, 6, 9, 0, 5, 3, 5, 0, 0, 7, 9, 1, 1, 7, 9, 9, 3, 5, 6, 1, 9, 5

Offset: 1

Author

Tomohiro Yamada, Jan 29 2019

Keywords

• nonn
• cons

Comments

This is equal to Product_{q>=1} (1-1/2^q)^(-1) over all q with 2^q - 1 a Mersenne prime.

Examples

			Decimal expansion of (4/3) * (8/7) * (32/31) * (128/127) * (8192/8191) * (131072/131071) * (524288/524287) * ... = 1.585558887...
		

Links

• Tomohiro Yamada, Table of n, a(n) for n = 1..99
• Tomohiro Yamada, Unitary super perfect numbers, Mathematica Pannonica, Volume 19, No. 1, 2008, pp. 37-47, using this constant with only a rough upper bound (4/3)*exp(4/21) < 1.6131008.

Crossrefs

Cf. A065446 (the corresponding product over all Mersenne numbers, prime or composite).

Cf. A173898 (the sum of reciprocals of the Mersenne primes).

Cf. A065442 (the sum of reciprocals of the Mersenne numbers, prime or composite).

Cf. A046528.

Programs

PARI

t=1.0;for(i=1,500,p=2^i-1;if(isprime(p),t=t*(p+1)/p))

Formula

Equals Sum_{n>=1} 1/A046528(n). - Amiram Eldar, Jan 06 2021