Professor
Anthony Begay of Navajo Community College influenced me in writing this
paper. I enjoyed the Smarandache concatenation. The sequences shown
here have been extracted from the Arizona State University(Tempe) Archives.
They are defined as follows:
(1) Smarandache Concatenated natural sequence:
1, 22, 333, 4444, 55555, 666666, 7777777,
88888888, 999999999, 10101010101010101010, 1111111111111111111111,
121212121212121212121212, 13131313131313131313131313, 1414141414141414141414141414,
151515151515151515151515151515, . . .
(2) Smarandache Concatenated prime sequence:
2, 23, 235, 2357, 235711, 23571113,
2357111317, 235711131719, 23571113171923, ...
Conjecture: there are infinitely many
primes among these numbers!
(3) Smarandache Concatenated odd sequence:
1, 13, 135, 1357, 13579, 1357911, 135791113,
13579111315, 1357911131517, ...
Conjecture: there are infinitely many
primes among these numbers!
(4) Smarandache Concatenated even sequence:
2, 24, 246, 2468, 246810, 24681012,
2468101214, 246810121416, ...
Conjecture: none of them is a perfect
power!
(5) Smarandache Concatenated S-sequence { generalization}:
Let s_{1}, s_{2},
s_{3}, s_{4}, . . . , s_{n}, . . . be an infinite
integer sequence (noted by S).
Then:
___ | ____ | ______ | _________ | |||
s_{1}, | s_{1}s _{2}, | s_{1}s_{2}s_{3}, | s_{1}s_{2}s_{3}s_{4}, | . . ., | s_{1}s_{2}s_{3}s_{4}...s _{n}, | . . . |
is called the Concatenated S-sequence.
Questions: (a) How many terms
of the Concatenated S-sequence belong to the initial
S-sequence?
(b) Or, how many terms of the
Concatenated S-sequence verify the
relation
of other given sequences?
The first three cases are particular.
Look now at some other examples, when
S is a sequence of squares, cubes, Fibonacci
respectively (and one can go so on).
(6) Smarandache Concatenated Square sequence:
1, 14, 149, 14916, 1491625, 149162536,
14916253649, 1491625364964, ...
How many of them are perfect squares?
(7) Smarandache Concatenated Cubic sequence:
1, 18, 1827, 182764, 182764125, 182764125216,
182764125216343, ...
How many of them are perfect cubes?
(8) Smarandache Concatenated Fibonacci sequence:
1, 11, 112, 1123, 11235, 112358, 11235813,
1123581321, 112358132134, ...
Does any of these numbers is a Fibonacci
number?
1. Smarandache, F. (1997). "Collected Papers", Vol. II,
University of Kishinev.
2. Smarandache, F. (1975). "Properties of the Numbers", University
of Craiova Archives.
[See also Arizona State University Special Collections,
Tempe, Arizona, USA].
* This paper originally appeared in Bulletin of Pure and Applied Sciences,
Vol. 16 E(No.2)
1997; p. 225-226.