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Extra info for A set of new Smarandache functions, sequences and conjectures in number theory
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? Problem 16. Evaluate for some Smarandache sequence a(n): ∞ ∑ n=1 Su (n ) ⋅ ln(a (n)) a (n) Is it convergent? 19) Smarandache Mertens Function Analogously to the Mertens function in number theory  we can define the following function: n Sm (n ) = ∑ k =1 48 Su (k ) where Su(k) is the Smarandache Mobius function applied to any Smarandache sequence a(k).
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). This question is well formulated if the hypothesis that the Smarandache sequence a(n) converges to a g constant value is satisfied. Problem 26 . For any Smarandache sequence evaluate: 1 N N ∑ ∑ ∑ Ss (k ) k =1 lim k →∞ k lim k →∞ 1 Ss (k ) Ss(k ) k 26) Smarandache prime factors function The Smarandache prime factors is defined for any Smarandache sequence a(n) as: Spf(n)=r(a(n)) where r(a(n)) is the number of distinct prime factors of a(n) .
728831.... 5336... 615792.... 99318... 913745.... 754... 513624.... 5587..... 239704.... 3... Problem 18. Is there any Smarandache sequence a(k) like the cubic-product one such that SR is close to an integer? Notice also that the product: F = π 4⋅S can produce numbers almost integer. 728831.... 994765....... 615792.... 983738....... 913745.... 007069....... 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?