Theory

A set of new Smarandache functions, sequences and by Felice Russo

By Felice Russo

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Additional resources for A set of new Smarandache functions, sequences and conjectures in number theory

Example text

Problem 12. Is S(C) finite? 13) Smarandache primitive sequence A Smarandache sequence in which a term cannot be dived by any other ones. 44 Example: Smarandache prime-digital sequence, that is prime-digit numbers which are themselves prime: 2, 3, 5, 7, 23, 37, 53, ... [6] (ID=A019546) 14) Smarandache alternating series A series of the form ∑ (−1) k ⋅ a (k )−1 k where a(k) is any Smarandache sequence. As example the alternate series has been calculated for the Smarandache square-product sequence [4].

Problem 14. Is there any Smarandache sequence such that for a particular value of s, for instance s0 , Sb (s0 ) = K where K is the Catalan constant [5]? 47 18) Smarandache Mobius Function For any Smarandache sequence a(n) we can define the following function analogously to the Mobius function in number theory [5]: 0 Su(n)= - 1 (−1)k if a(n) has one or more repeated prime factors if a(n)=1 if a(n) is a product of k distinct primes Problem 15. For a given Smarandache sequence a(n) evaluate lim n→ ∞ ∑ n Su ( n) where s ∈ N ns Is there any Smarandache sequence a(n) such that its Smarandache Zeta function Sz(n) is related to this sum?

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 [2] 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 [2] 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?