A099724 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.

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, 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, 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

Offset

0,1

Comments

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).

Links

Table of n, a(n) for n=0..104.

Formula

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

Example

0.8389098275921641893276775933054282385511940359741848509222502937433...

Maple

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

Mathematica

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 *)

Prog

(PARI) suminf(i=1, (prime(i+1) - prime(i))/exp(i)) \\ Michel Marcus, May 26 2018

Crossrefs

Cf. A000040, A001223, A001113.

Sequence in context: A368008 A085011 A250129 * A024568 A166202 A146482

Adjacent sequences: A099721 A099722 A099723 * A099725 A099726 A099727

Keyword

cons,nonn

Author

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

Status

approved