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A064648
Decimal expansion of sum of reciprocals of primorial numbers: 1/2 + 1/6 + 1/30 + 1/210 + ...
16
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
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.
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).
Sequence in context: A271177 A299624 A033150 * A116198 A137915 A316167
KEYWORD
cons,nonn
AUTHOR
Labos Elemer, Oct 04 2001
STATUS
approved