(OEIS A091588), where
(OEIS A085548) is the prime zeta function (Finch 2003). The coefficients and are offered by the sums
(Diaconis 1976, Knuth 2000, Diaconis 2002, Finch 2003, Knuth 2003), where
If is a primorial, then
(Hardy and also Wright 1979, p.355).
The summatory role of is offered by
where is the Mertens consistent (Hardy 1999, p.57), the ax (Hardy and also Ramanujan 1917; Hardy and Wright 1979, p.355) has been rewritten in a much more explicit form, and also and space asymptotic notation. The first few values that the summatory role are 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 15, 17, 19, 20, 21, ... (OEIS A013939). In addition,
(Hardy and also Wright 1979, p.357).
The first few numbers which are products of an odd number of distinct prime determinants (Hardy 1999, p.64; Ramanujan 2000, pp.xxiv and also 21) space 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 42, 43, 47, ... (OEIS A030059). satisfies
(Hardy 1999, pp.64-65). In addition, if is the variety of with REFERENCES:
Abramowitz, M. And also Stegun, I.A. (Eds.). Handbook the Mathematical attributes with Formulas, Graphs, and also Mathematical Tables, 9th printing. New York: Dover, p.844, 1972.
Diaconis, P. "Asymptotic Expansions because that the Mean and Variance that the number of Prime determinants of a Number ." Dept. Statistics Tech. Report 96, Stanford, CA: Stanford University, 1976.
Diaconis, P. "G.H.Hardy and also Probability???" Bull. LondonMath. Soc. 34, 385-402, 2002.
Finch, S. "Two Asymptotic Series." December 10, 2003. Http://algo.inria.fr/bsolve/.
Hardy, G.H. Ramanujan: Twelve Lectures ~ above Subjects suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.
Hardy, G.H. And Ramanujan, S. "The Normal variety of Prime determinants of a Number ." Quart. J. Math. 48, 76-92, 1917.
Hardy, G.H. And Wright, E.M. "The number of Prime factors of " and "The regular Order the and ." §22.10 and 22.11 in An arrival to the theory of Numbers, fifth ed. Oxford, England: Clarendon Press, pp.354-358, 1979.
Kac, M. Statistical freedom in Probability, analysis and Number Theory. Washington, DC: Math. Assoc. Amer., p.64, 1959.
Knuth, D.E. Selected papers on evaluation of Algorithms. Stanford, CA: CSLI Publications, pp.338-339, 2000.
Knuth, D.E. "Asymptotics for and ." quote by Finch (2003). Unpublished note, 2003.
Ramanujan, S. Accumulated Papers of Srinivasa Ramanujan (Ed. G.H.Hardy, P.V.S.Aiyar, and also B.M.Wilson). Providence, RI: Amer. Math. Soc., 2000.
Sloane, N.J.A. Order A001221/M0056, A013939, A027748, A085548, and also A091588 in "The On-Line Encyclopedia of integer Sequences."
Sloane, N.J.A. And Plouffe, S. TheEncyclopedia of creature Sequences. San Diego, CA: academic Press, 1995.
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