login
Decimal expansion of Sum_{k>0} (A000040(k+1)-A000040(k))/exp(k), where A000040(k) gives the k-th prime number and exp(k) is the natural exponential of k.
0

%I #15 May 26 2018 08:20:09

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

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

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

%N Decimal expansion of Sum_{k>0} (A000040(k+1)-A000040(k))/exp(k), where A000040(k) gives the k-th prime number and exp(k) is the natural exponential of k.

%C Relates the growth of a function giving the difference between successive prime numbers (A000040(n+1)-A000040(n) or A001223(n)) to the growth of the natural exponential exp(n)=e^n where e is Euler's number (A001113).

%F Equals Sum_{i >= 1} (prime(i+1) - prime(i))/exp(i).

%e 0.8389098275921641893276775933054282385511940359741848509222502937433...

%p f:=N->sum((ithprime(n+1)-ithprime(n))/exp(n),n=1..N); evalf[106](f(1000)); evalf[106](f(2000));

%t digits = 105; f[m_] := f[m] = Sum[(Prime[n + 1] - Prime[n])/Exp[n], {n, 1, m}] // RealDigits[#, 10, digits] & // First; f[digits]; f[m = 2*digits]; While[f[m] != f[m/2], m = 2 m]; f[m] (* _Jean-François Alcover_, Feb 21 2014 *)

%o (PARI) suminf(i=1, (prime(i+1) - prime(i))/exp(i)) \\ _Michel Marcus_, May 26 2018

%Y Cf. A000040, A001223, A001113.

%K cons,nonn

%O 0,1

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