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

0,1

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

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.

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

Example

0.705230171791800965147431682888248513743577639109154328192267913813919...

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 *)

Prog

(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

Crossrefs

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

Cf. A249270, A340469.

Sequence in context: A271177 A299624 A033150 * A116198 A137915 A316167

Adjacent sequences: A064645 A064646 A064647 * A064649 A064650 A064651

Keyword

cons,nonn

Author

Labos Elemer, Oct 04 2001

Status

approved