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Extra info for A set of new Smarandache functions, sequences and conjectures in number theory
A (n + 1) n → ∞ a( n ) • Evaluate lim where a(n) is the n-th term of the sequence. 18) Smarandache nn m generalized sequence The sequence is obtained concatenating n and n m with m ≥ 1 . 19) Smarandache n2n sequence 12, 24, 36, 48, 510, 1122, 1326, 1428, 1530, 1632, 1734, 1836, 1938 ...... For each n, the n-th term of the sequence is formed concatenating n and 2 ⋅ n . The n-th term of sequence is given by number of digits of 2n. a (n) = 2 ⋅ n + n ⋅ 10 d (2n) where d(2n) is the • How many terms in that sequence are perfect squares?
Is this sequence a Smarandache A and C sequence (see chapter II for definition)? • Let N+ be the number of terms even and N- the number of terms odd. Evaluate N− lim n→∞ N + 59 7) Smarandache alternate consecutive and reverse generalized sequence Let a(n) be a sequence with n ≥ 1 . The Smarandache alternate and reverse generalized sequence is given by: -------- -----------------------------a(1), a(2)a(1), a(1)a(2)a(3), a(4)a(3)a(2)a(1) ........ namely by the alternate concatenation of terms of a(n).
The even ones are always the consecutive primes but reversed, namely starting with 2 on the right. • How many terms are primes? • Note that the sum of digits of some term is a prime. How many terms are prime and the sum of their digits is prime too? • We define as “additive primes” a prime number whose digits sum is prime too (see sequence #11). • Is there any perfect square among the terms of that sequence? A perfect cube? 58 5) Smarandache alternate consecutive and reverse palprimes sequence (SACRPP) 2, 32, 235, 7532, 235711, 101117532, 235711101121, 131121101117532, 235711101121131151, 181151131121101117532, ....