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A064648
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Decimal expansion of sum of reciprocals of primorial numbers: 1/2 + 1/6 + 1/30 + 1/210 + ...
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16
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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
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OFFSET
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0,1
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COMMENTS
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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
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REFERENCES
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Friedrich Engel, "Entwicklung der Zahlen nach Stammbruechen" Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg. pp. 190-191, 1913.
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LINKS
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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.
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FORMULA
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(1/2)*(1 + (1/3)*(1 + (1/5)*(1 + (1/7)*(1 + (1/11)*(1 + (1/13)*(1 + ...)))))). - Jonathan Sondow, Aug 04 2014
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EXAMPLE
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0.705230171791800965147431682888248513743577639109154328192267913813919...
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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