By Felice Russo
ISBN-10: 1879585839
ISBN-13: 9781879585836
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Extra info for A set of new Smarandache functions, sequences and conjectures in number theory
Example text
Problem 24. Study E (Sd ( n)) for any Smarandache sequence where E(Sd(n)) is the n average value of Smarandache divisor function. 154.... 25) Smarandache Summatory Divisor function For any Smarandache sequence a(n) the following function can be defined: Ss (n) = σ(a (n )) where σ (a (n)) is the sum of the divisors of a(n) [5]. As example see the fig. 6. The function has been applied to the power sequence [2]. 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.
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? Problem 21. Is there any s ∈ N such that for a particular Smarandache sequence Se(s), Sl(s) and Sb(s) are all related to π ?
Since the product of Zw (n + 1) ⋅ Zw(n + 2) is always even if n is odd, it cannot be equal to n. Also if n, n+1 and n+2 are all squarefree then the equation has no solution. 3 we should have n = n 2 + 3n + 2 that of course is an absurd. Problem 5. Solve the equation Zw (n) ⋅ Zw (n + 1) = Zw( n + 2) Also in this case no solution has been found for the first 1000 values of Zw(n). Is is this true for all values of n? As for the problem 4, also in this case if n is odd the equation cannot have any solution.
A set of new Smarandache functions, sequences and conjectures in number theory by Felice Russo
by Charles
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