A184084 -id:A184084 - cp's OEIS frontend

A098640 a(n) = 2^p + 1 where p is the n-th prime.

5, 9, 33, 129, 2049, 8193, 131073, 524289, 8388609, 536870913, 2147483649, 137438953473, 2199023255553, 8796093022209, 140737488355329, 9007199254740993, 576460752303423489, 2305843009213693953

Offset: 1

Parthasarathy Nambi, Oct 27 2004

			If p=2, 2^2+1 = 5.
If p=7, 2^7+1 = 129.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A184084.

[2^p+1: p in PrimesUpTo(100)]; // Vincenzo Librandi, Apr 29 2014

Table[2^Prime[n] + 1, {n, 18}] (* Robert G. Wilson v, Nov 01 2004 *)

a(n) = 2^prime(n) + 1; \\ Amiram Eldar, Nov 22 2022

Product_{n>=1} (1 - 1/a(n)) = A184084. - Amiram Eldar, Nov 22 2022

More terms from Robert G. Wilson v, Nov 01 2004

A184083 Decimal expansion of product_{p=primes} (1+1/2^p).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jan 09 2011

Inverse of the constant A184084.

			(1+1/2^2) *(1+1/2^3) *(1+1/2^5) *(1+1/2^7) *(1+1/2^11)* .... = 1.4624312190069598013...

RealDigits[Times@@(1+1/2^Prime[Range[100]]),10,120][[1]] (* Harvey P. Dale, Sep 28 2013 *)

prodeuler(p=2,1e6,1.+(1.>>p)) \\ Charles R Greathouse IV, Aug 09 2022

Equals product_{p in A000040} (1+2^(-p)) = sum_{n>=0} A000586(n)/2^n.

A293258 Decimal expansion of product of 1 - 4^-p over all primes p.

Original entry on oeis.org

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

Offset: 0

Charles R Greathouse IV, Oct 04 2017

Knopfmacher proves that prime(n+1) = floor(1 - log(1 - A/P)) where A is this constant and P is the product (1 - 4^-2)(1 - 4^-3)(1 - 4^-5)...(1 - 4^-prime(n)).

			0.921893835296931859194670302799807186732205478738862674976230660393864453122...

John Knopfmacher, Recursive formulae for prime numbers, Archiv der Mathematik (Basel) 33:2 (1979/80), pp. 144-149.

Cf. A184082, A184083, A184084.

prodeuler(p=2, bitprecision(1.)/2+2, 1-4.^-p)

Equals A184082 * A184083 = A184082 / A184084. - Amiram Eldar, Nov 16 2021

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A098640 a(n) = 2^p + 1 where p is the n-th prime.

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A184083 Decimal expansion of product_{p=primes} (1+1/2^p).

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A293258 Decimal expansion of product of 1 - 4^-p over all primes p.

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PARI

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