By Felice Russo
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Extra info for A set of new Smarandache functions, sequences and conjectures in number theory
1, Zw(p)=p. But according to the result of Bach and Shallit , the sum of first N primes is asymptotically equal to: N2 2 ⋅ ln( N ) and this completes the proof. 1 The difference |Zw(n+1)-Zw(n)| is unbounded. k k Let’s suppose that n = 2 2 for k ≥ 0 . Then n + 1 = 2 2 + 1 , namely (n+1) is a Fermat number . If the Lehmer & Schinzel conjecture is true  then every Fermat number is squarefree. 5 and then this difference can be as large as we want. 33 Of course if the Lehmer & Schinzel conjecture is true, conjecture will be true.
Problem 10. Find all values of n such that Zw(Z(n))-Z(Zw(n))>0 Problem 11. Find all values of n such that Zw(Z(n))-Z(Zw(n))<0 Problem 12. Study the functions Zw(Z(n)), Z(Zw(n)) and Zw(Z(n))-Z(Zw(n)). Problem 13. 12325672.... where the sequence of digits is Zw(n) for n ≥ 1 an irrational or trascendental number? (We call this number the PseudoSmarandache-Squarefree constant). Problem 14. Is the Smarandache Euler-Mascheroni sum (see chapter II for definition) convergent for Zw(n) numbers? If yes evaluate the convergence value.
For a given Smarandache sequence a(n) evaluate ∞ ∑ St (n) n=1 ns where s ∈ N Is there any Smarandache sequence such that its Smarandache zeta function Sz(n) is related to this sum? 24) Smarandache divisor function 52 The Smarandache divisor function is defined as the number of the divisors d of a(n), where a(n) is any Smarandache sequence, that is: Sd(n)=d(a(n)) As example see fig. 5 where the function Sd has been applied to the Smarandache power sequence . Smarandache divisor function for Smarandache power sequence 18 16 14 Sd(n) 12 10 8 6 4 2 0 1 34 67 100 133 166 199 232 265 298 331 364 397 430 463 496 n Fig.
A set of new Smarandache functions, sequences and conjectures in number theory by Felice Russo