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properties of the numerical function fs

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PROPERTIES OF THE l'4'lJMERICAL FliNCTION s by I. Blilcenoiu, V. Seleacu, N. Virlan Departament of Mathematics, University of Craiova Craiova (1100), ROMANIA In this paper are studied some properties o f the numerical function Fs(x):N - {O,I} 4 N Fs(x) = L Sp(x), where 5 p (x) = S(p ) is the Smarandache O<pS.r p pnme function defined in [4]. Numerical example: Fs(5) = S(2 5 ) ~ 5 5 ) S(5 5 ); Fs(6) = S( 2 6 ) 5 6 ) ( ~ ) . It is known that: ( p -1)r 1 5(pr) pr so(p -1)r < S(pr) pr. Than (1) Where Jr x) is the number of prime numbers smaller or equal with x. . % 1 PROPOSITION 1: The sequence T( r = 1 log Fs (x) + L . has limit - 00. i=2 F:(l) Proof The inequality Fs(x) r(P2+···4- P:n..r - Jr(x» implies -logFs(x) < < -logr(A. P2 . ·+P1I(%) - n(r» < -logr(n(r)A. - ;r(r» = -logr -log 1Z (x) -log(A. -1). Than for x=i the inequality (1) become : 1 1 1 1 < < = FS i) i(Pl + '+P:(I) - Jr(i» i(PIJr(i) - Jr(i» iJr(i)(Pt -1 .r 1 Than T(x)<l-log(x)-logJr(x)-log(A-l)+L . . - .=2 m(I)(Pt -1) .r 1 A = 2 =: T(x) = l-logx-logJr(x)+ L . . 1=2 lJr(l) =: lim T(x) 1 - lim logx-lim 10gJr(x) lim i: . _1_. =1-oo-oo+L=-oo. ;c~ ;c .... ;c .... ;c .... oc 1=2 I Jr I PRoPOsmON 1. The equation Fs x) = Fs x + 1) has no solution for x eN - to I}. 6

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