By Felice Russo
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Additional resources for A set of new Smarandache functions, sequences and conjectures in number theory
64 20) Smarandache nkn generalized sequence with k>1 For each n, the n-th term of sequence is formed concatenating together n and k ⋅ n . ---- ---- ---- ---- ---1k1, 2k2, 3k3, 4k4, 5k5, .................. 21) Smarandache multiple sequence 1, 24, 369, 481216, 510152025, 61218243036, 7142128354249, 816243240485664 The nth term is obtained concatenating together n, 2 ⋅ n, 3 ⋅ n, K K n ⋅ n . • Evaluate for that sequence the simple continued fraction and the continued radical. • How many terms are primes?
Is that sequence finite? 9) Smarandache additive Square sequence 1, 4, 9, 36, 81, 100, 121, 144, 169, 196, 225, 324, 400, 441, 484, 529, 900, 961, … Sequence of squares which digits sum is a square too. • Is this sequence finite? • Which is the maximum number of consecutive square that are also additive square? (In the first terms reported above this maximum number is 7). 60 10) Smarandache additive Cubic sequence 1, 8, 125, 512, 1000, 1331, 8000, 19683, 35937, ......... Sequence of cubes which digits sum is a cube too.
Today the largest known sequence of this type contains 15 terms and start with the prime 2442113). 61 12) Smarandache multiplicative square sequence 1, 4, 9, 49, 144, 289, ......... Sequence of squares which product of digits is a square too. • Is that sequence finite? • How many terms in that sequence are also additive squares? 13) Smarandache multiplicative Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, ........ Sequence of Fibonacci numbers which digit's product is a Fibonacci number too. • Which is the maximum number of consecutive Fibonacci numbers that are also Fibonacci-additive?
A set of new Smarandache functions, sequences and conjectures in number theory by Felice Russo