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By Felice Russo

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

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4). Smarandache Mertens Function Power sequence 30 25 20 Sm(n) 15 10 5 0 -5 -10 1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106 113 120 127 134 141 148 155 162 169 176 183 190 197 n Mertens function sqrt(a(n)) Fig. 4 20) Smarandache Ramanujan Constant For any Smarandache sequence a(n) we can define the following function: 4 ⋅S SR = eπ ⋅ int(π ) where S = ∑ 1 and assuming that S converge. 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?

S ⋅ t − 1) But s ⋅ t − p ⋅ q = 1 being n and n+1 consecutive. This implies that = k but s ⋅t this is absurd because s ⋅ t − 1 and s ⋅ t don't have any common factor. So our initial assumption is false. Same thing for the equation 31 Zw( n + 1) =k. Zw(n) For case 2 without loss of generality let’s suppose that: n = p a ⋅ q b and n + 1 = sc ⋅ t d where p, q, s, t are distinct primes. 2, Zw (n) = p ⋅ q and Zw (n + 1) = s ⋅ t and then let’s s ⋅t suppose that =k. p⋅q Since s ⋅ t − p ⋅ q = 1 , being n and n+1 consecutive, m ⋅ s ⋅ t − m' ⋅ p ⋅ q = 1 where m and m' are two integers, because s c ⋅ t d and p a ⋅ q b are multiple of s ⋅ t and p ⋅ q respectively.

1, 1998, p. 149-151 [4] F. Russo, Some results about four Smarandache U-product sequences, Smarandache Notions Journal, Nov. 1999 [5] E. Weisstein, CRC Concise Encyclopedia of Mathematics, CRC Press, 1999 [6] N. com/ ~njas/sequences/ 56 Chapter III A set of new Smarandache sequences. The greatest discovery of all humankind may well have been the natural numbers. For many past aeons, philosophers and mathematicians have studied the sequence of natural numbers, uncovering startling and mystifying truths.