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%I #17 Apr 30 2020 04:57:10
%S 5,1,6,9,4,2,8,1,9,8,0,5,6,4,0,3,8,4,2,4,0,5,1,6,6,0,8,4,7,9,8,5,6,2,
%T 7,7,9,7,8,5,4,6,9,4,7,9,1,3,0,9,1,2,4,1,6,5,0,2,8,0,2,4,5,8,7,1,2,3,
%U 8,0,7,5,3,4,1,1,3,6,0,3,7,7,1,9,8,1,8,0,2,8,0,5,4,0,2,5,0,8,8,2
%N Decimal expansion of Sum_{p prime} 1/(2^p-1), a prime equivalent of the Erdős-Borwein constant.
%C Erdős was interested in the question whether this constant is irrational. - _Amiram Eldar_, Apr 30 2020
%D Paul Erdős, Some of my favourite unsolved problems, in A. Baker, B. Bollobás and A. Hajnal (eds.), A Tribute to Paul Erdős, Cambridge University Press, 1990, p. 470.
%H Paul Erdős, <a href="https://users.renyi.hu/~p_erdos/1948-04.pdf">On arithmetical properties of Lambert series</a>, J. Indian Math. Soc. (N.S.), Vol. 12 (1948), pp. 63-66.
%H Paul Erdős, <a href="https://users.renyi.hu/~p_erdos/1969-09.pdf">On the irrationality of certain series</a>, Math. Student, Vol. 36 (1969), pp. 222-226.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Erdos-BorweinConstant.html">Erdős-Borwein Constant</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeConstant.html">Prime Constant</a>.
%F Equals Sum_{i>=1} 1/A001348(i). - _R. J. Mathar_, Feb 17 2016
%F Equals Sum_{k>=1} omega(k)/2^k, where omega(k) is the number of distinct primes dividing k (A001221). - _Amiram Eldar_, Apr 30 2020
%e 0.51694281980564038424051660847985627797854694791309124165028...
%t digits = 100; m0 = 100; dm = 100; s[m_] := s[m] = N[Sum[1/(2^Prime[n]-1), {n, 1, m}], digits+10]; s[m = m0]; Print[{m, s[m]}]; s[m = m + dm]; While[ Print[{m, s[m]}]; RealDigits[ s[m], 10, digits+5] != RealDigits[ s[m - dm], 10, digits+5], m = m + dm]; RealDigits[ s[m], 10, digits] // First
%o (PARI) suminf(k=1, omega(k)/2^k) \\ _Michel Marcus_, Apr 30 2020
%Y Cf. A051006, A065442.
%K nonn,cons
%O 0,1
%A _Jean-François Alcover_, Sep 13 2015
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