A184083 -id:A184083 - cp's OEIS frontend

A034785 a(n) = 2^(n-th prime).

4, 8, 32, 128, 2048, 8192, 131072, 524288, 8388608, 536870912, 2147483648, 137438953472, 2199023255552, 8796093022208, 140737488355328, 9007199254740992, 576460752303423488, 2305843009213693952

Offset: 1

Asher Auel

These are the "outputs" in Conway's PRIMEGAME (see A007542). - Alonso del Arte, Jan 03 2011

Multiplicative encoding of the n-th prime. - Daniel Forgues, Feb 26 2017

			a(4) = 128 because the fourth prime number is 7 and 2^7 = 128.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index to divisibility sequences.

Cf. A000040, A000430, A051006, A073718 (2^(n-th composite)), A074736.

Cf. A184082, A184083.

a034785 = (2 ^) . a000040
-- Reinhard Zumkeller, Feb 07 2015, Jan 24 2012

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

2^Prime@Range@40  (* Vladimir Joseph Stephan Orlovsky, Apr 11 2011 *)

a(n)=1<Charles R Greathouse IV, Apr 07 2012

from sympy import prime
def A034785(n): return 1<Chai Wah Wu, Aug 09 2024

From Amiram Eldar, Aug 11 2020: (Start)
a(n) = 2^A000040(n).
Sum_{n>=1} 1/a(n) = A051006. (End)
From Amiram Eldar, Nov 22 2022: (Start)
Product_{n>=1} (1 + 1/a(n)) = A184083.
Product_{n>=1} (1 - 1/a(n)) = A184082. (End)

More terms from James Sellers, Feb 04 2000

A184084 Decimal expansion of product_{p=primes} (1-1/(2^p+1)).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 09 2011

Inverse of the constant A184083.

			(1-1/5) *(1-1/9) *(1-1/33) * (1-1/129) *(1-1/2049)* ... = 0.6837928423594743689943636...

RealDigits[Times@@Table[1-1/(2^p+1),{p,Prime[Range[1000]]}],10,100][[1]] (* Harvey P. Dale, Jul 23 2024 *)

Equals product_{p in A000040} (1-1/(2^p+1)) = 1/ product_p (1+2^(-p)) = product_{n>=1} (1-1/A098640(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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A034785 a(n) = 2^(n-th prime).

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A184084 Decimal expansion of product_{p=primes} (1-1/(2^p+1)).

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Comments

Examples

Programs

Mathematica

Formula

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

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula