A249270 - OEIS

A249270

Decimal expansion of lim_{n->oo} (1/n)*Sum_{k=1..n} smallest prime not dividing k.

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The old definition was "Decimal expansion of the mean value over all positive integers of the least prime not dividing a given integer."

The integer parts of the sequence having this constant as starting value and thereafter x[n+1] = (frac(x[n])+1)*floor(x[n]), where floor and frac are integer and fractional part, are exactly the sequence of the prime numbers: see the Grime-Haran Numberphile video for details. - M. F. Hasler, Nov 28 2020

REFERENCES

Steven R. Finch, Meissel-Mertens constants: Quadratic residues, Mathematical Constants, Cambridge Univ. Press, 2003, pp. 96—98.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..10000

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

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 171.

Dylan Fridman, Juli Garbulsky, Bruno Glecer, James Grime and Massi Tron Florentin, A Prime-Representing Constant, The American Mathematical Monthly, Vol. 126, No. 1 (2019), pp. 70-73, ResearchGate link.

James Grime and Brady Haran, 2.920050977316, Numberphile video, Nov 26 2020.

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.

Juan L. Varona, A Couple of Transcendental Prime-Representing Constants, arXiv:2012.11750 [math.NT], 2020.

I. A. Weinstein, Family of prime-representing constants: use of the ceiling function, arXiv:2101.00094 [math.GM], 2021.

FORMULA

Sum_{k >= 1} (p_k - 1)/(p_1 p_2 ... p_{k-1}), where p_k is the k-th prime number.

Sum_{k >= 0} 1/A034386(k). - Jani Melik, Jul 22 2015

From Amiram Eldar, Oct 29 2020: (Start)

Equals lim_{n->oo} (1/n) * Sum_{k=1..n} A053669(k).

Equals 2 + Sum_{n>=1} (prime(n+1)-prime(n))/prime(n)# = 2 + Sum_{n>=1} A001223(n)/A002110(n). (End)

prime(n+1) = floor(C*prime(n)# - prime(n)*floor(C*prime(n-1)# - 1)) with prime(1)=2 where C is this constant. - Davide Rotondo, Sep 15 2023

EXAMPLE

2.9200509773161347120925629171120194680027278993214267...

MATHEMATICA

digits = 103; Clear[s]; s[m_] := s[m] = Sum[(Prime[k] - 1)/Product[Prime[j], {j, 1, k - 1}] // N[#, digits + 100]&, {k, 1, m}]; s[10]; s[m = 20]; While[RealDigits[s[m]] != RealDigits[s[m/2]], m = 2*m]; RealDigits[s[m], 10, digits] // First

PROG

(Sage)

def sharp_primorial(n): return sloane.A002110(prime_pi(n));

@CachedFunction

def spv(n):

b = 0

for i in (0..n):

b += 1 / sharp_primorial(i)

return b

N(spv(300), digits=108) # Jani Melik, Jul 22 2015

CROSSREFS

Cf. A000040, A001223, A002110, A034386, A053669, A098990, A232927.