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A064648 Decimal expansion of sum of reciprocals of primorial numbers: (1/2)+(1/6)+(1/30)+(1/210)+... = 0.7052301717918009651474316828882485137435776391... 10
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

This is the real number whose Engel expansion is precisely the sequence of primes A000040. - Jonathan Vos Post, May 04 2005

The sum of the reciprocals of the primorial numbers is an irrational number, by Cantor's 1869 theorem on the irrationality of Cantor series. - Jonathan Sondow, Jun 22 2012

REFERENCES

G. Cantor, On simple number systems (Ueber die einfachen Zahlensysteme), Z. Math. und Phys. 14 (1869), 121-128; also in "Collected Works", Springer-Verlag, Berlin, 1932, pp. 35-42.

Engel, F. "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

Simon Plouffe, Sum of the product of prime reciprocals

Eric Weisstein's World of Mathematics, Engel Expansion.

EXAMPLE

0.705230171791800965147431682888248513743577639109154328192267913813919... [From Harry J. Smith, Sep 21 2009]

MAPLE

A064648:=proc(q)

local a, b, n;

a:=1; b:=0;

for n from 1 to q do  a:=a*ithprime(n); b:=b+1/a; od; print(evalf(b, 100));

end:

A064648(1000); # Paolo P. Lava, Jan 15 2013.

MATHEMATICA

RealDigits[ Sum[1/Product[ Prime[i], {i, n}], {n, 58}], 10, 111][[1]] (from Robert G. Wilson v, Aug 05 2005)

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

CROSSREFS

Cf. A002110.

Cf. A054543, A000027, A053977, A006784, A028259.

Cf. A165509 (continued fraction).

Sequence in context: A021997 A099737 A033150 * A116198 A137915 A198114

Adjacent sequences:  A064645 A064646 A064647 * A064649 A064650 A064651

KEYWORD

cons,nonn

AUTHOR

Labos Elemer, Oct 04 2001

STATUS

approved

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Last modified April 19 08:09 EDT 2014. Contains 240739 sequences.