 (5)
where (OEIS A091588), where (8)
(OEIS A085548) is the prime zeta role (Finch 2003). The coefficients and also are given by the sums(Diaconis 1976, Knuth 2000, Diaconis 2002, Finch 2003, Knuth 2003), where (Finch 2003). If is a primorial, then (18)
(Hardy and Wright 1979, p.355). The summatory function of is provided by (19)
where is the Mertens consistent (Hardy 1999, p.57), the hatchet (Hardy and Ramanujan 1917; Hardy and Wright 1979, p.355) has been rewritten in a an ext explicit form, and also and space asymptotic notation. The first few values the the summatory duty are 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 15, 17, 19, 20, 21, ... (OEIS A013939). In addition, (20)
(Hardy and also Wright 1979, p.357). The first couple of numbers which are products of one odd variety of distinct prime components (Hardy 1999, p.64; Ramanujan 2000, pp.xxiv and also 21) are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 42, 43, 47, ... (OEIS A030059). satisfies (21)
(Hardy 1999, pp.6465). In addition, if is the variety of with REFERENCES:Abramowitz, M. And Stegun, I.A. (Eds.). Handbook of Mathematical functions with Formulas, Graphs, and also Mathematical Tables, ninth printing. New York: Dover, p.844, 1972. Diaconis, P. "Asymptotic Expansions for the Mean and Variance that the number of Prime components 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, 385402, 2002. Finch, S. "Two Asymptotic Series." December 10, 2003. Http://algo.inria.fr/bsolve/. Hardy, G.H. Ramanujan: Twelve Lectures on Subjects suggested by His Life and also Work, third ed. Brandnew York: Chelsea, 1999. Hardy, G.H. And also Ramanujan, S. "The Normal variety of Prime determinants of a Number ." Quart. J. Math. 48, 7692, 1917.Hardy, G.H. And also Wright, E.M. "The number of Prime components of " and also "The common Order that and ." §22.10 and 22.11 in An advent to the concept of Numbers, 5th ed. Oxford, England: Clarendon Press, pp.354358, 1979.Kac, M. Statistical selfreliance in Probability, evaluation and Number Theory. Washington, DC: Math. Assoc. Amer., p.64, 1959. Knuth, D.E. Selected files on analysis of Algorithms. Stanford, CA: CSLI Publications, pp.338339, 2000. Knuth, D.E. "Asymptotics because that and ." quote by Finch (2003). Unpublished note, 2003.Ramanujan, S. Gathered Papers that Srinivasa Ramanujan (Ed.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 OnLine Encyclopedia of creature Sequences." Sloane, N.J.A. And Plouffe, S. TheEncyclopedia of essence Sequences. San Diego, CA: scholastic Press, 1995. Referenced on sdrta.netAlpha: distinct Prime Factors
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