Department of Mathematics,
Searching through the Archives of the 
I found interesting sequences of numbers and problems 
related to them. I display some of them, and the readers 
are welcome to contribute with solutions or ideas. 
Key words:
Smarandache Pdigital subsequences, Smarandache Ppartial subsequences, 
Smarandache type partition, Smarandache Ssequences, 
Smarandache uniform sequences, Smarandache operation sequences. 
Let { a_{n} }, n > 1 be a sequence
defined by a property (or a relationship involving its terms P.) Now, we screen
this sequence, selecting only its terms whose digits hold the property (or
relationship involving the digits) P.
The new sequence obtained is called:
(1) Smarandache Pdigital subsequences.
For example:
(a) Smarandache squaredigital subsequence:
0, 1, 4, 9, 49, 100, 144, 400, 441, . . .
i.e. from 0, 1, 4, 9, 16, 25, 36, ..., n^{2}, ... we choose only
the terms whose digits are all perfect squares
(therefore only 0, 1, 4, and 9).
Disregarding the square numbers of the form
___________ 
N0 . . . 0, 
2k zeros 
where N is also a perfect square, how many
other numbers belong to this sequence?
(b) Smarandache cubedigital subsequence:
0, 1, 8, 1000, 8000, . . .
i.e. from 0, 1, 8, 27, 64, 125, 216, . . . , n^{3}, . . . we choose
only the terms whose digits are all perfect cubes
(therefore only 0, 1 and 8).
Similar question, disregarding the cube numbers of the form
___________ 
M0 . . . 0, 
3k zeros 
where M is a perfect cube.
(c) Smarandache prime digital subsequence:
2, 3, 5, 7, 23, 37, 53, 73, . . .
i.e. the prime numbers whose digits are all primes.
Conjecture: this sequence is infinite.
In the same general conditions of a given sequence, we screen it selecting only
its terms whose groups of digits hold the property (or relationship involving
the groups of digits) P.
[ A group of digits may contain one or more digits, but not the whole term.]
The new sequence obtained is called:
(2) Smarandache Ppartial digital subsequence.
Similar examples:
(a) Smarandache squarepartialdigital subsequence:
49, 100, 144, 169, 361, 400, 441, . . .
i.e. the square members that is to be partitioned into groups of digits which
are also perfect squares.
(169 can be partitioned as 16 = 4^{2} and 9 = 3^{2}, etc.)
Disregarding the square numbers of the form
___________ 
N0 . . . 0, 
2k zeros 
where N is also a perfect square, how many
other numbers belong to this sequence?
(b) Smarandache cubepartial digital subsequence:
1000, 8000, 10648, 27000, . . .
i.e. the cube numbers that can be partitioned into groups of digits which are
also perfect cubes.
(10648 can be partitioned as 1 = 1^{3}, 0 = 0^{3}, 64 = 4^{3},
and 8 = 2^{3}).
Same question: disregarding the cube numbers of the form:
___________ 
M0 . . . 0, 
3k zeros 
where M is also a perfect cube, how many
other numbers belong to this sequence.
(c) Smarandache primepartial digital subsequence:
23, 37, 53, 73, 113, 137, 173, 193, 197, . . .
i.e. prime numbers, that can be partitioned into groups of digits which are
also prime,
(113 can be partitioned as 11 and 3, both primes).
Conjecture: this sequence is infinite.
(d) Smarandache Lucaspartial digital sunsequence
123, . . .
i.e. the sum of the two first groups of digits is equal to the last group of
digits, and the whole number belongs to Lucas numbers:
2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, . . .
(beginning at 2 and L(n+2) = L(n+1) + L(n), n > 1) ( 123 is partitioned as
1, 2 and 3, then 3 = 2 + 1). Is 123 the only Lucas number that verifies a
Smarandache type partition?
Study some Smarandache P  (partial)  digital subsequences associated to:
 Fibonacci numbers (we were not able to find any Fibonacci number verifying a
Smarandache type partition, but we could not investigate large numbers; can
you? Do you think none of them would belong to a Smarandache F  partial
digital subsequence?
 Smith numbers, Eulerian numbers, Bernoulli numbers,
Mock theta numbers, Smarandache type sequences etc.
Remark: Some sequences may not be smarandachely
partitioned (i.e. their associated Smarandache type subsequences are empty).
If a sequence {a_{n} }, n >= 1 is defined by a_{n} = f(n) (
a function of n), then Smarandache fdigital subsequence is obtained by
screening the sequence and selecting only its terms that can be partitioned in
two groups of digits g_{1} and g_{2} such that
g_{2} = f(g_{1} ).
For example:
(a) If a_{n} = 2n, n >= 1, then
Smarandache evendigital subsequence is:
12, 24, 36, 48, 510, 612, 714, 816, 918, 1020, 1122, 1224, . . .
(i.e. 714 can be partitioned as g_{1} = 7, g_{2} = 14, such
that 14 = 2*7, etc. )
(b) Smarandache luckydigital subsequence
37, 49, . . .
(i.e. 37 can be partitioned as 3 and 7, and L_{3} = 7; the lucky
numbers are
1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, . . .
How many other numbers belong to this subsequence? Study the Smarandache
fdigital subsequence associated to other wellknown sequences.
(3) Smarandache odd sequence:
1, 3, 135, 1357, 13579, 1357911, 135791113, 13579111315, 1357911131517, ...
How many of them are prime?
(4) Smarandache even sequence:
2, 24, 246, 2468, 246810, 24681012, 2468101214, 246810121416, . . .
Conjecture: None of them is a perfect power!
(5) Smarandache prime sequence:
2, 23, 235, 2357, 235711, 23571113, 2357111317, 235711131719,
23571113171923, . . .
How many of them are prime?
(Conjecture: a finite number).
(6) Smarandache Ssequence:
General definition:
Let S_{1} , S_{2} , S_{3} , . . . , 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_{n} 
. . 
is called the Smarandache Ssequence.
Question:
(a) How many of the Smarandache Ssequence belong
to the initial S sequence?
(b) Or, how many of the Smarandache Ssequence verify the relation of other
given sequences?
For example:
If S is the sequence of odd numbers 1, 3, 5, 7, 9, . . . then the Smarandache
Ssequence is 1, 13, 135, 1357, . . . [(i.e.1)] and all the other terms are
odd;
Same if S is the sequence of even numbers [(i.e. 2)]
The question (a) is trivial in this case.
But,when S is the sequence
of primes [i.e. 3], the question becomes much harder.
Study the case when S (replaced by F) is the Fibonacci sequence (for one
example):
1, 1, 2, 3, 5, 8, 13, 21, . . . .
Then the Smarandache F  sequence
1, 11, 112, 1123, 11235, 112358, . . .
How many primes does it contain?
(7) Smarandache uniform sequences:
General definition:
Let n be an integer not equal to zero and d_{1} , d_{2} , . . .
, d_{r} digits in a base B
(of course r < B).
Then: multiples of n, written with digits d_{1} ,
d_{2} , . . . ,d_{r} only (but all r
of them), in base B, increasingly ordered, are called the Smarandache uniform
sequence.
As a particular case it's important to study the multiples written with one
digit only (when r = 1).
Some examples (in base 10):
(a) Multiples of 7 written with digit 1 only:
111111, 111111,111111, 111111,111111,111111, 111111,111111,111111,111111, ...
(b) Multiples of 7 written with digit 2 only:
222222, 222222222222, 222222222222222222, 222222222222222222222222, ...
(c) Multiples of 79365 written with digit 5 only:
555555, 555555555555, 555555555555555555, 555555555555555555555555, ...
For some cases, the Smarandache uniform sequence may be empty (impossible):
(d) Multiples of 79365 written with digit 6 only (because any multiple of 79365
will end in 0 or 5.
Remark: If there exists at least a multiple m of n, written with digits d_{1}
,d_{2} , . . ., d_{r} only, in base
B, then there exists an infinite number of multiples of n (they have the form:

___ 
____ 
_____ 
m, 
mm, 
mmm, 
mmmm, . . . 
With a computer program it's easy to select all multiples (written with
certain digits) of a given number  up to some limit.
Exercise: Find the general term expression for multiples of 7 written with digits
1, 3, 5 only in base 10.
(8) Smarandache operation sequences:
General definition:
Let E be an ordered set of elements, E = { e_{1} ,e_{2} , . . .
} and O a set of binary operations welldefined for these elements.
Then: a_{1} is an element of { e_{1}
,e_{2} , . . . }.
a_{n+}_{1} = min { e_{1} O_{1}
e_{2} O_{2} . . . O_{n} e_{n+1} } > a_{n}
, for n > 1.
where all O_{i} are
operations belonging to O, is called the Smarandache operation sequence.
Some examples:
(a) When E is the natural number set, and O is formed by the four arithmetic operations:
+, , *, /.
Then: a_{1} = 1
a_{n+1} = min { 1 O_{1} 2 O_{2}
. . . O_{n} (n+1) } > a_{n} , for n > 1,
(therefore, all O_{i} may be chosen among
addition, subtraction, multiplication or division in a convenient way).
Questions: Find this Smarandache arithmetics operation
infinite sequence. Is it possible to get a general expression formula for
this sequence (which starts with 1, 2, 3, 5, 4,?
(b) A finite sequence
a_{1} = 1
a_{n+1} = min { 1 O_{1} 2 O_{2}
... O_{98} 99 } > a_{n}
for n > 1, where all O_{i} are elements of
{ +, , *, / }.
Same questions for this Smarandache arithmetics
operation finite sequence.
(c) Similarly for Smarandache algebraic operation infinite sequence
a_{1} = 1
a_{n+1} = min { 1 O_{1} 2 O_{2} . . . O_{n}
(n+1) } > a_{n} for n > 1,
where all O_{i} are elements of { +, , *, /,
**, ysqrtx }
( X**Y means X^{Y} and ysqrtx means the yth root of x).
The same questions become harder but more exciting.
(d) Similarly for Smarandache algebraic operation finite sequence:
a_{1} = 1
a_{n+1} = min { 1 O_{1} 2 O_{2} . . . O_{98}
99} > a_{n} , for n > 1,
where all O_{i} are elements of { +, , *, /,
**, ysqrtx }
( X**Y means X^{Y} and ysqrtx means the yth root of x).
Same questions.
More generally: one replaces "binary operations" by "K_{i} ary
operations"
where all K_{i} are integers >= 2).
Therefore, a_{i} is an element of
{ e_{1} , e_{2} , . . . },
a_{n+}_{1}
= min{ 1 O_{1}^{(K}_{1}^{)}2O_{1}^{(K}_{1}^{)}
. . . O_{1}^{(}^{K}_{1}^{)} K_{1}
O_{2}^{(K}_{2}^{)}(K_{2}+1O_{2}^{(K2)}
. . . O_{2}^{(} 
O_{1}^{(K}_{1}^{)}
is K_{1}  ary
O_{2}^{(K}_{2}^{)} is 


(n+2K_{r})O_{r}^{(}^{K}_{r}^{)} . . . O_{r}^{(K}_{r}^{)}(n+1)} > a_{n}, for n > 1 
Of course K_{1} + (K_{2}  1) + . . .+
(K_{r}  1) = n+1.
Remark: The questions are much easier when O = { +,};
study the Smarandache operation type sequences in this case.
(9) Smarandache operation sequences at random:
Same definitions and questions as for the previous sequences, except that
a_{n+1} = { e_{1} O_{1} e_{2} O_{2} . .
. O_{n} e_{n+1} } > a_{n} , for n > 1,
(i.e. it's no "min" any more, therefore a_{n+1} will
be chosen at random, but greater than a_{n} , for any n > 1). Study
these sequences with a computer program for random variables (under weak
conditions).
1. Smarandache, F. (1975) "Properties of the Numbers", Arhivele Statului,
Filiala Valcea, Rm. Valcea, Romania,
[see also
* Originally appeared in Bulletin of Pure and Applied Sciences, Vol. 15 E (No. 1), 1996; pp. 101107.