 
SMARANDACHE CONCATENATE TYPE SEQUENCES*
H. Marimutha
Northland Pioneer College (USA)
ABSTRACT
Professor
A. 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 Ssequence { 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 Ssequence.
Questions: (a) How many terms
of the Concatenated Ssequence belong to the initial
Ssequence?
(b) Or, how many terms of the
Concatenated Ssequence 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?
REFERENCES
1. Smarandache, F. (1997). "Collected Papers", Vol. II,
University of Kishinev.
2. Smarandache, F. (1975). "Properties of the Numbers", Arhivele Statului Valcea,
Rm. Valcea, Romania, 1972;
[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; pp. 225226.
 
