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.

References

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.2.1, p. 96.

Links

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

Thomas Bloom, Problem 251, Erdős Problems.

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.

Erdős problems database contributors, Erdős problem database, see no. 251.

Steven 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: A198457 A388160 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