2212 Sacele,
Some Smarandache relationships between the terms of a given sequence |
are studied in the first paragraph. In the second paragraph, are studied |
Smarandache subsequences (whose terms have the same property as the |
initial sequence). In the third paragraph are studied the Smarandache |
magic squares and cubes of order n and some conjectures in number |
theory. |
Key Words:
Smarandache p-q relationships, Smarandache p-q-<>-subsequence, |
Smarandache type subsequences, Smarandache type partition, Smarandache |
type definitions, Smarandache type conjectures in number theory. |
Let { a_{n}
}, n >= 1 be a sequence of numbers and p, q integers greater than or equal
to 1. Then we say that the terms
a_{k+1},a_{k+2} , . . . , a_{k+p},a_{k+p+1},a_{k+p+2}
,. . ., a_{k+p+q}_{
}verify a Smarandache p-q relationship if
a_{k+1} <> a_{k+2} <> . . . <> a_{k+p} = a_{k+p+1} <> a_{k+p+2}
<> . . . <> a_{k+p+q}_{
}where "<>" may be any arithmetic or algebraic or analytic
operation (generally a binary law on { a_{1} , a_{2} , a_{3}
, . . . }).
If this relationship is verified for any k >= 1 (i.e. by all terms of the
sequence), then
{ a_{n} }, n >= 1 is called a Smarandache p-q-<> sequence
where "<>" is replaced by "additive" if <>
= +, "multiplicative" if <> = *, etc. [according to the
operation (<>) used].
As a [particular case, we can easily see that Fibonacci/Lucas sequence (a_{n}
+ a_{n+1} = a_{n+2 )}, for n >= 1 is a Smarandache 2-1 additive
sequence.
A Tribonacci sequence ( a_{n}
+ a_{n+1} + a_{n+2} = a_{n+3} ), n >= 1 is a
Smarandache 3-1 additive sequence. Etc.
Now, if we consider the sequence of Smarandache numbers,
1, 2, 3, 4, 5, 3, 7, 4, 6, 5, 11, 4, 13, 7, 5, 6, 17, . . . ,
i.e. for each n the smallest number S(n) such that S(n)! is
divisible by n[See(1)] (the values of the Smarandache Function), it raises the
questions:
(a) How many quadruplets verify a Smarandache 2-2 additive relationship i.e.
S(n+1) + S(n+2) = S(n+3) + S(n+4)?
I found: S(6) + S(7) = S(8) + S(9), 3 + 7 = 4 + 6;
S(7) + S(8)
= S(9) + S(10), 7 + 4 = 6 + 5;
S(28) +
S(29) = S(30) + S(31), 7 + 29 = 5 + 31.
But, what about others? I am not able to tell you if
there exist a finite or infinite number (?)
(b) How many quadruplets verify a Smarandache 2-2-subtractive relationship,
i.e.
S(n+1) - S(n+2) = S(n+3)
- S(n+4)?
I found: S(1) - S(2) = S(3) - S(4), 1 - 2 = 3 - 4;
S(2) - S(3)
= S(4) - S(5), 2 - 3 = 4 - 5;
S(49) -
S(50) = S(51) - S(52), 14 - 10 = 17 - 13.
(c) How many sextuplets verify a Smarandache 3-3 additive relationship, i.e.
S(n+1) + S(n+2) + S(n+3)
= S(n+4) + S(n+5) + S(n+6)?
I found: S(5) + S(6) + S(7) = S(8) + S(9) + S(10), 5 +
3 + 7 = 4 + 6 + 5.
I read that Charles Ashbacher has a computer program
that calculates the Smarandache Function's values,
therefore he may be able to add more solutions to mine.
More generally:
If f_{p} is a p-ary
relation and g_{q} is a q-ary
relation, both of them defined on { a_{1} ,a_{2} ,a_{3}
, . . . }, then
a_{i1} ,a_{i2} , . . . , a_{ip} , a_{j1} ,a_{j2} , . . . ,a_{jq}_{
}verify a Smarandache f_{p} - g_{q} - relationship if
f_{p} (a_{i1} ,a_{i2} , . . .
,a_{ip} ) = g_{q}
(a_{j1} ,a_{j2} , . . . ,a_{jq}
).
If this relationship is verified by all terms of the sequence, then {a_{n} }, n >= 1 is called a Smarandache f_{p} -g_{q}
-sequence.
Study some Smarandache f_{p} -g_{q} - relationships for
well-known sequences (perfect numbers, Ulam numbers,
abundant numbers, Catalan numbers, Cullen numbers, etc.).
For example: a Smarandache 2-2-additive, subtractive, or multiplicative
relationship, etc.
If f_{p} is a p-ary
relation on { a_{1} ,a_{2} ,a_{3} , . . . } and
f_{p} (a_{i1} ,a_{i2} , . . .
,a_{ip} ) = f_{p}a_{j1} , (a_{j2}
, . . . ,a_{jp} )
for all a_{ik} ,a_{jk}
, where k = 1, 2, . . . ,p, and for all p >= 1, then {a_{n} }, n
>= 1, is called a Smarandache perfect f sequence.
If not all p-plets (a_{i1} , a_{i2} ,
. . . ,a_{ip} ) and (a_{j1} ,a_{j2}
, . . . , a_{jp} ) verify the f_{p} relation, or not for all p >= 1,
the relation f_{p} is verified, then
{a_{n} }, n >= 1 is called a Smarandache partial perfect
f-sequence.
An example: a Smarandache partial perfect additive sequence:
1, 1, 0, 2, -1, 1, 1, 3, -2, 0, 0, 2, 1, 1, 3, 5, -4, -2, -1, 1, -1, 1, 1, 3,
0, 2, . . .
This sequence has the property that
a_{1} + a_{2} + . . . + a_{p} = a_{p+1} + a_{p+2}
+ . . . + a_{2p} for all p >= 1.
It is constructed in the following way:
a_{1} = a_{2} = 1
a_{2p+1} = a_{p+}_{1}
- 1
a_{2p+2} = a_{p+1} + 1
for all p >= 1.
(a) Can you, readers, find a general expression of a_{n} (as function
of n)?
It is periodical, or convergent or bounded?
(b) Please design (invent) yourselves other Smarandache perfect (or partial
perfect) sequences.
Think about a multiplicative sequence of this type.
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 those digits hold the
property (or relationship involving the digits) P.
The new sequence obtained is called:
(1) Smarandache P-digital subsequence:
For example:
(a) Smarandache square-digital 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 cube-digital 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 P-partial digital subsequence.
Similar examples:
(a) Smarandache square-partial-digital 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 cube-partial 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 prime-partial 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 Lucas-partial digital subsequence
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 f-digital 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 even-digital 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 lucky-digital 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, . . .
For n >= 2, let A be a set of n^{2}
elements, and l an n-ary law defined on A.
As a generalization of the XVIth - XVIIth centuries magic squares, we present the Smarandache
magic square of order n which is a square array of rows of elements of A
arranged so that the law l applied to each horizontal and vertical row
and diagonal gives the same result.
If A is an arithmetic progression and l the addition
of n numbers, then many magic squares have been found. Look at Durer's 15/4 engraving "Melancholia's" one:
16 3
2 13
5 10
11 8
9 6
7 12
4 15
14 1
(1) Can you find such a magic square of order at least 3 or 5, when A is a set
of prime numbers and l the addition?
(2) Same question when A is a set of square numbers, or cube numbers, or
special numbers [for example: Fibonacci or Lucas numbers, triangular numbers,
Smarandache quotients (i.e. q(m) is the smallest k
such that mk is a factorial), etc.].
A similar definition for the Smarandache magic cube of order n, where
the elements of A are arranged in the form of a cube of length n:
(a) either each element inside of a unitary cube (that the initial cube is
divided in).
(b) either each element on a surface of a unitary
cube.
(c) either each element on a vertex of a unitary cube.
(3) Study similar questions for this case, which is more complex.
An interesting law may be
l(a_{1} ,a_{2} , . . . ,a_{n})
= a_{1} + a_{2} - a_{3} + a_{4} - a_{5}
+ . . .
Smarandache prime conjecture:
Any odd number can be expressed as the sum of two primes minus a third prime
(not including the trivial solution
p = p + q - q when the odd number is the prime itself).
For example:
1 = 3 + 5 - 7 = 5 + 7 - 11 = 7 + 11 - 17 = 11 + 13 - 24 = . . .
3 = 5 + 11 - 13 = 7 + 19 - 23 = 17 + 23 - 37 = . . .
5 = 3 + 13 - 11 = . . .
7 = 11 + 13 - 17 = . . .
9 = 5 + 7 - 3 = . . .
11 = 7 + 17 - 13 = . . .
(a) Is this conjecture equivalent to Goldbach's
conjecture (any odd number > = 9 can be expressed as a sum of three primes -
finally solved 15 by Vinogradov in 1937 for any odd
number greater than _{3}3^{15}
)?
(b) The number of times each odd number can be
expressed as a sum of two primes minus a third prime are called Smarandache
prime conjecture numbers. None of them are known!
(c) Write a computer program to check this conjecture for as many positive odd
numbers as possible.
(2) There are infinitely many numbers that cannot be expressed as the
difference between a cube and a square (in absolute value).
They are called Smarandache bad numbers(!)
For example: 5, 6, 7, 10, 13, 14, . . . are probably such bad numbers (F.
Smarandache has conjectured, see[1]), while
1, 2, 3, 4, 8, 9, 11, 12, 15, . . . are not, because
1 = | 2^{3} - 3^{2} |
2 = | 3^{3} - 5^{2} |
3 = | 1^{3} - 2^{2} |
4 = | 5^{3} - 11^{2} |
8 = | 1^{3} - 3^{2} |
9 = | 6^{3} - 15^{2} |
11 = | 3^{3} - 4^{2} |
12 = | 13^{3} - 47^{2} |
15 = | 4^{3} - 7^{2} |, etc.
(a) Write a computer program to get as many non Smarandache bad numbers (it's
easier this way!) as possible,
i.e. find an ordered array of a's such that
a = | x^{3} - y^{2} |, for x and y integers >= 1.
1. Smarandache, F. (1975). "Properties
of Numbers", University of
2. Sloane, N. J. A. and Simon, Plouffe, (1995). The Encyclopedia of Integer Sequences, Academic Press,
* This paper first appeared in Bulletin of Pure and Applied Sciences, Vol. 17E
(No.1) 1998; p. 55-62.