A033150 Decimal expansion of Niven's constant.

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, 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, 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

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• cons

Comments

This constant is the average value of A051903. - Charles R Greathouse IV, Oct 30 2012

There are no 9's in the first 50 digits after the decimal point. Then, suddenly, it goes 909999. - Bobby Jacobs, Aug 13 2017

Named after the Canadian-American mathematician Ivan Morton Niven (1915 - 1999). - Amiram Eldar, Aug 19 2020

Examples

			1.7052111401...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, pp. 112-115.

Links

• C. W. Anderson, Problem 6015, The American Mathematical Monthly, Vol. 82, No. 2 (1975), pp. 183-184, T. Salat, Prime Decomposition of Integers, solution to Problem 6015, ibid., Vol. 83, No. 10 (1976), p. 820.
• Ivan Niven, Averages of Exponents in Factoring Integers, Proc. Amer. Math. Soc., Vol. 22, No. 2 (1969), pp. 356-360.
• Simon Plouffe, The Niven constant to 256 digits.
• Kaneenika Sinha, Average orders of certain arithmetical functions, Journal of the Ramanujan Mathematical Society, Vol. 21, No. 3 (2006), pp. 267-277.
• Eric Weisstein's World of Mathematics, Niven's Constant.
• Wikipedia, Niven's constant.

Crossrefs

Cf. A008683, A033151, A033152, A033153, A033154.

Programs

Mathematica

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 *)

PARI

1+suminf(j=2,1-1/zeta(j)) \\ Charles R Greathouse IV, Aug 13 2017

Formula

Equals 1 + Sum_{j>=2} 1-(1/zeta(j)).

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

Extensions

Offset corrected by Oleg Marichev (oleg(AT)wolfram.com), Jan 28 2008