A098990 Decimal expansion of Sum_{n>=1} prime(n)/(2^n).

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, 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, 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

Offset

1,1

Comments

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

Links

Table of n, a(n) for n=1..105.

Peter J. Cho and Henry H. Kim, The average of the smallest prime in a conjugacy class, International Mathematics Research Notices, Vol. 2020, No. 6 (2020), pp. 1718-1747, arXiv preprint, arXiv:1601.03012 [math.NT], 2016.

Paul Erdős, Remarks on number theory. I., Mat. Lapok, Vol. 12 (1961), pp. 10-17; Math. Rev. 26 #2410.

S. R. Finch, Average least nonresidues, December 4, 2013. [Cached copy, with permission of the author]

Paul Pollack, The average least quadratic nonresidue modulo m and other variations on a theme of Erdős, J. Number Theory, Vol. 132, No. 6 (2012), pp. 1185-1202, alternative link.

Formula

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

Equals 2 plus the constant in A098882. - R. J. Mathar, Sep 02 2008

Equals lim_{n->oo} (1/n) * Sum_{k=1..n} A053760(k). - Amiram Eldar, Oct 29 2020

Example

3.6746439660113287789956763090840294116777975887794373283122052201763...

Maple

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

Mathematica

RealDigits[Sum[Prime[i]/2^i, {i, 1000}], 10, 120][[1]] (* Harvey P. Dale, Apr 10 2012 *)

Prog

(PARI) suminf(k=1, prime(k)/2^k) \\ Michel Marcus, Jan 13 2016

Crossrefs

Cf. A000040, A053760, A098882.

Sequence in context: A243976 A198457 A264921 * A336453 A162195 A117361

Adjacent sequences: A098987 A098988 A098989 * A098991 A098992 A098993

Keyword

cons,nonn

Author

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

Status

approved