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%I #87 Oct 09 2023 12:03:37
%S 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,
%T 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,
%U 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
%N Decimal expansion of lim_{n->oo} (1/n)*Sum_{k=1..n} smallest prime not dividing k.
%C The old definition was "Decimal expansion of the mean value over all positive integers of the least prime not dividing a given integer."
%C 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
%D Steven R. Finch, Meissel-Mertens constants: Quadratic residues, Mathematical Constants, Cambridge Univ. Press, 2003, pp. 96—98.
%H Alois P. Heinz, <a href="/A249270/b249270.txt">Table of n, a(n) for n = 1..10000</a>
%H Steven R. Finch, <a href="/A232927/a232927.pdf">Average least nonresidues</a>, December 4, 2013. [Cached copy, with permission of the author]
%H Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 171.
%H Dylan Fridman, Juli Garbulsky, Bruno Glecer, James Grime and Massi Tron Florentin, <a href="https://doi.org/10.1080/00029890.2019.1530554">A Prime-Representing Constant</a>, The American Mathematical Monthly, Vol. 126, No. 1 (2019), pp. 70-73, <a href="https://www.researchgate.net/publication/330746181_A_Prime-Representing_Constant">ResearchGate link</a>.
%H James Grime and Brady Haran, <a href="https://youtube.com/watch?v=_gCKX6VMvmU">2.920050977316</a>, Numberphile video, Nov 26 2020.
%H Paul Pollack, <a href="http://dx.doi.org/10.1016/j.jnt.2011.12.015">The average least quadratic nonresidue modulo m and other variations on a theme of Erdős</a>, J. Number Theory, Vol. 132, No. 6 (2012), pp. 1185-1202.
%H Juan L. Varona, <a href="https://arxiv.org/abs/2012.11750">A Couple of Transcendental Prime-Representing Constants</a>, arXiv:2012.11750 [math.NT], 2020.
%H I. A. Weinstein, <a href="https://arxiv.org/abs/2101.00094">Family of prime-representing constants: use of the ceiling function</a>, arXiv:2101.00094 [math.GM], 2021.
%F Sum_{k >= 1} (p_k - 1)/(p_1 p_2 ... p_{k-1}), where p_k is the k-th prime number.
%F Sum_{k >= 0} 1/A034386(k). - _Jani Melik_, Jul 22 2015
%F From _Amiram Eldar_, Oct 29 2020: (Start)
%F Equals lim_{n->oo} (1/n) * Sum_{k=1..n} A053669(k).
%F Equals 2 + Sum_{n>=1} (prime(n+1)-prime(n))/prime(n)# = 2 + Sum_{n>=1} A001223(n)/A002110(n). (End)
%F 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
%e 2.9200509773161347120925629171120194680027278993214267...
%t 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
%o (Sage)
%o def sharp_primorial(n): return sloane.A002110(prime_pi(n));
%o @CachedFunction
%o def spv(n):
%o b = 0
%o for i in (0..n):
%o b += 1 / sharp_primorial(i)
%o return b
%o N(spv(300), digits=108) # _Jani Melik_, Jul 22 2015
%Y Cf. A000040, A001223, A002110, A034386, A053669, A098990, A232927.
%K nonn,cons
%O 1,1
%A _Jean-François Alcover_, Oct 24 2014
%E Definition revised by _N. J. A. Sloane_, Nov 29 2020
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