Felice Russo's A set of new Smarandache functions, sequences and PDF
By Felice Russo
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Additional resources for A set of new Smarandache functions, sequences and conjectures in number theory
6. The function has been applied to the power sequence . 97 91 85 79 73 67 61 55 49 43 37 31 25 19 13 7 180 160 140 120 100 80 60 40 20 0 1 Ss(n) Smarandache Summatory Divisor Function for Smarandache power sequence n Fig. 3275 K Problem 25. Is there any Smarandache sequence a(n) such that the following inequality is satisfied? 2 ⋅ (1 − 2 ) + g − ln(4 ⋅ π ) Ss(n ) ≤ eg + a( n) ⋅ ln(ln( a( n))) ln(a (n)) ⋅ ln(ln( a(n ))) where g is the Smarandache Euler-Mascheroni constant for the sequence a(n).
20 The diophantine equation Zw(n)=Zw(n+1) has no solutions. Proof. In fact according to the previous theorem if n is even (odd respectively) then Zw(n) also is even (odd respectively). Therefore the equation Zw(n)=Zw(n+1) can not be satisfied because Zw(n) that is even should be equal to Zw(n+1) that instead is odd. 21 Zw( n + 1) The equations =k Zw(n ) Zw (n + 1) and Zw( n) −1 =k with k any positive integer and n>1 for the first equation don't admit solutions. Proof. We must consider three cases: 1.
513624.... 031646....... 239704.... 75843....... If F is close to an integer we call it a Smarandache almost integer. Problem 19. Is there any Smarandache sequence a(n) that produce other Smarandache almost integers? 4 The Smarandache Dirichlet Eta function for the Smarandache power sequence  diverge for s<3. 22) Smarandache Dirichlet Lambda Function Let a(n) be any Smarandache sequence. 5 The Smarandache Dirichlet lambda function for the power sequence  diverge for s<5. Problem 20. Is there any Smarandache sequence a(n) such that: Sz ( s) + Se( s) = 2 ⋅ Sl (s ) for some s ∈ N , where Sz(s) is the Smarandache Zeta function and Se(s) the Smarandache Dirichlet Eta function?
A set of new Smarandache functions, sequences and conjectures in number theory by Felice Russo