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%I #35 Aug 19 2020 06:48:08
%S 1,7,0,5,2,1,1,1,4,0,1,0,5,3,6,7,7,6,4,2,8,8,5,5,1,4,5,3,4,3,4,5,0,8,
%T 1,6,0,7,6,2,0,2,7,6,5,1,6,5,3,4,6,9,0,9,9,9,9,4,2,8,4,9,0,6,5,4,7,3,
%U 1,3,1,9,2,1,6,8,1,2,2,4,9,1,9,3,4,2,4,4,1,3,2,1,0,0,8,7,1,0,0,1,7,9
%N Decimal expansion of Niven's constant.
%C This constant is the average value of A051903. - _Charles R Greathouse IV_, Oct 30 2012
%C There are no 9's in the first 50 digits after the decimal point. Then, suddenly, it goes 909999. - _Bobby Jacobs_, Aug 13 2017
%C Named after the Canadian-American mathematician Ivan Morton Niven (1915 - 1999). - _Amiram Eldar_, Aug 19 2020
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, pp. 112-115.
%H C. W. Anderson, <a href="http://www.jstor.org/stable/2319672">Problem 6015</a>, The American Mathematical Monthly, Vol. 82, No. 2 (1975), pp. 183-184, T. Salat, <a href="http://www.jstor.org/stable/2318702">Prime Decomposition of Integers, solution to Problem 6015</a>, ibid., Vol. 83, No. 10 (1976), p. 820.
%H Ivan Niven, <a href="https://doi.org/10.1090/S0002-9939-1969-0241373-5">Averages of Exponents in Factoring Integers</a>, Proc. Amer. Math. Soc., Vol. 22, No. 2 (1969), pp. 356-360.
%H Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/niven.txt">The Niven constant to 256 digits</a>.
%H Kaneenika Sinha, <a href="https://www.iiserkol.ac.in/~kaneenika/maxnrevfinal.pdf">Average orders of certain arithmetical functions</a>, Journal of the Ramanujan Mathematical Society, Vol. 21, No. 3 (2006), pp. 267-277.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NivensConstant.html">Niven's Constant</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Niven%27s_constant">Niven's constant</a>.
%F Equals 1 + Sum_{j>=2} 1-(1/zeta(j)).
%F Equals 1 - Sum_{k>=2} mu(k)/(k*(k-1)), where mu is the Möbius function (A008683) (Anderson, 1975; Sinha, 2006). - _Amiram Eldar_, Aug 19 2020
%e 1.7052111401...
%t rd[n_] := rd[n] = RealDigits[ N[1 + Sum[1 - 1/Zeta[j], {j, 2, 2^n}] , 105]][[1]]; rd[n = 4]; While[rd[n] =!= rd[n-1], n++]; rd[n] (* _Jean-François Alcover_, Oct 25 2012 *)
%o (PARI) 1+suminf(j=2,1-1/zeta(j)) \\ _Charles R Greathouse IV_, Aug 13 2017
%Y Cf. A008683, A033151, A033152, A033153, A033154.
%K nonn,cons
%O 1,2
%A _Eric W. Weisstein_
%E Offset corrected by Oleg Marichev (oleg(AT)wolfram.com), Jan 28 2008
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