A064648 Decimal expansion of sum of reciprocals of primorial numbers: 1/2 + 1/6 + 1/30 + 1/210 + ...

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

Offset: 0

Author

Labos Elemer, Oct 04 2001

Keywords

• cons
• nonn

Comments

The Engel expansion of this constant is the sequence of primes. - Jonathan Vos Post, May 04 2005

Let S be the operator over the space omega of infinite sequences of numbers, defined to be the Engel expansion of the sum of reciprocals of primorials of a sequence p of numbers; than the eigenvalue-equation S p = p is satisfied by the sequence of prime numbers. - Ralf Steiner, Dec 31 2016

This constant is irrational (Griffiths, 2015). - Amiram Eldar, Oct 27 2020

Examples

			0.705230171791800965147431682888248513743577639109154328192267913813919...
		

References

• Friedrich Engel, "Entwicklung der Zahlen nach Stammbruechen" Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg. pp. 190-191, 1913.

Links

• Harry J. Smith, Table of n, a(n) for n = 0..20000
• Friedrich Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
• Martin Griffiths, 99.29 On the sum of the reciprocals of the primorials, The Mathematical Gazette, Vol. 99, No. 546 (2015), pp. 522-523.
• Simon Plouffe, Sum of the product of prime reciprocals.
• Eric Weisstein's World of Mathematics, Engel Expansion.

Crossrefs

Cf. A002110, A054543, A000027, A053977, A006784, A028259, A165509 (continued fraction).

Cf. A249270, A340469.

Programs

Mathematica

RealDigits[ Sum[1/Product[ Prime[i], {i, n}], {n, 58}], 10, 111][[1]] (* Robert G. Wilson v, Aug 05 2005 *)
RealDigits[Total[1/#&/@FoldList[Times,Prime[Range[100]]]],10,120][[1]] (* Harvey P. Dale, Aug 27 2019 *)

PARI

default(realprecision, 20080); p=1; s=x=0; for (k=1, 10^9, p*=prime(k); s+=1.0/p; if (s==x, break); x=s ); x*=10; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b064648.txt", n, " ", d)) \\ Harry J. Smith, Sep 21 2009

Sage

@CachedFunction
def pv(n):
    a = 1
    b = 0
    for i in (1..n):
        a *= nth_prime(i)
        b += 1/a
    return b
N(pv(100),digits=108) # From Maple code Jani Melik, Jul 22 2015

Formula

(1/2)*(1 + (1/3)*(1 + (1/5)*(1 + (1/7)*(1 + (1/11)*(1 + (1/13)*(1 + ...)))))). - Jonathan Sondow, Aug 04 2014

Equals Sum_{n>=1} 1/A002110(n). - Amiram Eldar, Oct 27 2020