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

%I A064648 #94 Feb 16 2025 08:32:45
%S A064648 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,
%T A064648 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,
%U A064648 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
%N A064648 Decimal expansion of sum of reciprocals of primorial numbers: 1/2 + 1/6 + 1/30 + 1/210 + ...
%C A064648 The Engel expansion of this constant is the sequence of primes. - _Jonathan Vos Post_, May 04 2005
%C A064648 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
%C A064648 This constant is irrational (Griffiths, 2015). - _Amiram Eldar_, Oct 27 2020
%D A064648 Friedrich Engel, "Entwicklung der Zahlen nach Stammbruechen" Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg. pp. 190-191, 1913.
%H A064648 Harry J. Smith, <a href="/A064648/b064648.txt">Table of n, a(n) for n = 0..20000</a>
%H A064648 Friedrich Engel, <a href="/A006784/a006784.pdf">Entwicklung der Zahlen nach Stammbruechen</a>, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
%H A064648 Martin Griffiths, <a href="https://doi.org/10.1017/mag.2015.91">99.29 On the sum of the reciprocals of the primorials</a>, The Mathematical Gazette, Vol. 99, No. 546 (2015), pp. 522-523.
%H A064648 Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/primeprod.txt">Sum of the product of prime reciprocals</a>.
%H A064648 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EngelExpansion.html">Engel Expansion</a>.
%F A064648 (1/2)*(1 + (1/3)*(1 + (1/5)*(1 + (1/7)*(1 + (1/11)*(1 + (1/13)*(1 + ...)))))). - _Jonathan Sondow_, Aug 04 2014
%F A064648 Equals Sum_{n>=1} 1/A002110(n). - _Amiram Eldar_, Oct 27 2020
%e A064648 0.705230171791800965147431682888248513743577639109154328192267913813919...
%t A064648 RealDigits[ Sum[1/Product[ Prime[i], {i, n}], {n, 58}], 10, 111][[1]] (* _Robert G. Wilson v_, Aug 05 2005 *)
%t A064648 RealDigits[Total[1/#&/@FoldList[Times,Prime[Range[100]]]],10,120][[1]] (* _Harvey P. Dale_, Aug 27 2019 *)
%o A064648 (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
%o A064648 (Sage)
%o A064648 @CachedFunction
%o A064648 def pv(n):
%o A064648     a = 1
%o A064648     b = 0
%o A064648     for i in (1..n):
%o A064648         a *= nth_prime(i)
%o A064648         b += 1/a
%o A064648     return b
%o A064648 N(pv(100),digits=108) # From Maple code _Jani Melik_, Jul 22 2015
%Y A064648 Cf. A002110, A054543, A000027, A053977, A006784, A028259, A165509 (continued fraction).
%Y A064648 Cf. A249270, A340469.
%K A064648 cons,nonn
%O A064648 0,1
%A A064648 _Labos Elemer_, Oct 04 2001