By Felice Russo
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Additional resources for A set of new Smarandache functions, sequences and conjectures in number theory
Problem 9. Evaluate: ∞ S ( B 2) = sup all Smaran. B 2 seq. ∑ k =1 1 b(k ) that is, the largest reciprocal sum for all Smarandache B2 sequences. 12) Smarandache nonaveraging sequence (or Smarandache C sequences) An infinite Smarandache sequence c(n) of positive integers 1 ≤ c(1) ≤ c( 2) ≤ c(3)K is said nonaveraging if it contains no three terms in arithmetic progression. That is, c(i) + c( j) ≠ 2 ⋅ c (k ) for any three distinct terms c(i), c(j) and c(k) forming the sequence. Problem 10. Find some examples of Smarandache nonaveraging sequence.
Each sequence is labeled with an increasing initial number. 1) Smarandache repeated digit sequence with 1-endpoints 111, 1221, 13331, 144441, 1555551, 16666661, 177777771, 1888888881, 19999999991, 1101010101010101010101,..... Starting with n repeat it n times and then add 1 on the left and on the rigth. • • How many terms are primes? Observe that the sum of digits of each term is given by n 2 + 2 . Then the n-th term will be not a prime if n2 + 2 is an integer. 3 57 2) Smarandache repeated digit generalized sequence with n-endpoints n1n, n22n, n333n, n4444n, n55555n, n666666n, n7777777n, n88888888n, ....
Conjecture: None. • How many terms of that sequence are squarefree? 17) Smarandache nn3 sequence 11, 28, 327, 464, 5125, 6216, 7343, 8512, 9729, 101000, 111331, 121728, 63 132197, 142744, 153375, 164096 ...... The sequence is formed concatening n and n3 for n=1, 2, 3, 4..... 3 The n-th term of the sequence is given by a (n) = n ⋅ 10 d ( n ) + n 3 where d ( n 3 ) is the number of digits of n3 . • How many terms in this sequence are palindromes? • How many primes are in that sequence? Conjecture: A finite number.
A set of new Smarandache functions, sequences and conjectures in number theory by Felice Russo