%I #36 Dec 03 2020 20:56:57

%S 3,6,7,4,6,4,3,9,6,6,0,1,1,3,2,8,7,7,8,9,9,5,6,7,6,3,0,9,0,8,4,0,2,9,

%T 4,1,1,6,7,7,7,9,7,5,8,8,7,7,9,4,3,7,3,2,8,3,1,2,2,0,5,2,2,0,1,7,6,3,

%U 7,9,8,6,7,0,4,4,8,2,8,3,6,0,4,1,7,4,5,4,7,6,4,5,7,8,8,0,1,9,0,1,1,3,7,5,2

%N Decimal expansion of Sum_{n>=1} prime(n)/(2^n).

%C Relates the growth of the n-th prime function A000040(n) to the base-2 exponential of n.

%H Peter J. Cho and Henry H. Kim, <a href="https://doi.org/10.1093/imrn/rny074">The average of the smallest prime in a conjugacy class</a>, International Mathematics Research Notices, Vol. 2020, No. 6 (2020), pp. 1718-1747, <a href="http://arxiv.org/abs/1601.03012">arXiv preprint</a>, arXiv:1601.03012 [math.NT], 2016.

%H Paul Erdős, <a href="http://www.renyi.hu/~p_erdos/1961-23.pdf">Remarks on number theory. I.</a>, Mat. Lapok, Vol. 12 (1961), pp. 10-17; Math. Rev. 26 #2410.

%H S. R. Finch, <a href="/A232927/a232927.pdf">Average least nonresidues</a>, December 4, 2013. [Cached copy, with permission of the author]

%H Paul Pollack, <a href="https://doi.org/10.1016/j.jnt.2011.12.015">The average least quadratic nonresidue modulo m and other variations on a theme of Erdős</a>, J. Number Theory, Vol. 132, No. 6 (2012), pp. 1185-1202, <a href="http://pollack.uga.edu/variations.pdf">alternative link</a>.

%F Equals Sum_{n>=1} prime(n)/2^n.

%F Equals 2 plus the constant in A098882. - _R. J. Mathar_, Sep 02 2008

%F Equals lim_{n->oo} (1/n) * Sum_{k=1..n} A053760(k). - _Amiram Eldar_, Oct 29 2020

%e 3.6746439660113287789956763090840294116777975887794373283122052201763...

%p f:=N->sum(ithprime(n)/2^n,n=1..N); evalf[106](f(500)); evalf[106](f(1000));

%t RealDigits[Sum[Prime[i]/2^i,{i,1000}],10,120][[1]] (* _Harvey P. Dale_, Apr 10 2012 *)

%o (PARI) suminf(k=1, prime(k)/2^k) \\ _Michel Marcus_, Jan 13 2016

%Y Cf. A000040, A053760, A098882.

%K cons,nonn

%O 1,1

%A Joseph Biberstine (jrbibers(AT)indiana.edu), Nov 07 2004