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Decimal expansion of the Sum_{n>0} (A000040(n+1)-A000040(n))/(2^n), where A000040(k) gives the k-th prime number.
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%I #14 Nov 17 2020 05:39:37

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

%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 the Sum_{n>0} (A000040(n+1)-A000040(n))/(2^n), where A000040(k) gives the k-th prime number.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeNumber.html">Prime Number</a>.

%F Equals A098990 - 2. - _Amiram Eldar_, Nov 17 2020

%e 1.6746439660113287789956763090840294116777975887794373283122052201763...

%p g:=N->sum((ithprime(n+1)-ithprime(n))/2^n,n=1..N); evalf[106](g(5000)); evalf[106](g(10000));

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

%Y Cf. A000040, A098990.

%K cons,nonn

%O 1,2

%A Joseph Biberstine (jrbibers(AT)indiana.edu), Nov 03 2004