SOME NOTIONS ON LEAST COMMON
MULTIPLES
(Amarnath Murthy, S.E. (E&T),Well Logging Services, Oil and Natural Gas corporation Ltd.,Sabarmati, Ahmedabad, 380 005 , INDIA.)
In [1] Smarandache LCM Sequence has been defined as T_{n} = LCM ( 1 to n ) = LCM of all the natural numbers up to n.
The SLS is
1, 2, 6, 60, 60, 420, 840, 2520, 2520, . . .
We denote the LCM of a set of numbers a, b, c, d, etc. as LCM(a,b,c,d)
We have the well known result that n! divides the product of any set of n consecutive numbers. Using this idea we define Smarandache LCM Ratio Sequence of the r^{th} kind as SLRS(r)
The n_{ }^{th }term _{r}T_{n} =LCM (n , n+1, n+2, . . .n+r-1 ) /LCM ( 1, 2, 3, 4, . . . r )
As per our definition we get SLRS(1) as
1 , 2, 3, 4, 5, . . . _{1}T_{n} (= n.)
we get SLRS(2) as
1, 3, 6, 10, . . . _{2}T_{n} = n(n+1)/2 ( triangular numbers).
we get SLRS(3) as
LCM (1, 2, 3)/ LCM (1, 2, 3), LCM (2, 3, 4 )/ LCM (1, 2, 3) , LCM ( 3, 4, 5,)/ LCM (1, 2, 3)
LCM (4, 5, 6)/ LCM (1, 2, 3) LCM (5, 6, 7)/ LCM (1, 2, 3)
=== 1 , 2 , 10 , 10 , 35 . . . similarly we have
SLRS(4) === 1, 5 , 5, 35, 70, 42, 210 , . . .
It can be noticed that for r > 2 the terms do not follow any visible patterns.
OPEN PROBLEM : To explore for patterns/ find reduction formullae for _{r}T_{n}
.
Definition: Like ^{n}C_{r , }the combination of r out of n given objects , We define a new term ^{n}L_{r}
As
^{n}L_{r} = LCM (
n, n-1, n-2, . . . n-r+1 ) / LCM ( 1, 2 , 3 , . . .r )
(Numeretor is the LCM of n , n-1 , n-2, . . .n-r+1 and the denominator is the LCM of first natural numbers.)
we get ^{1}L_{0} =1, ^{1}L_{1} =1,^{ 2}L_{0}
=1,^{ 2}L_{1} =2,^{ 2}L_{2} =2 etc. define ^{0}L_{0}
=1
we get the following triangle:
1 |
1 ,1 |
1 ,2, 1 |
1 ,3 ,3, 1 |
1, 4 ,6, 2, 1 |
1, 5, 10,, 10 5, 1 |
1, 6, 15, 10, 5, 1, 1 |
1, 7, 21, 35, 35, 7,7, 1 |
1, 8, 28, 28, 70, 14, 14, 2, 1 |
1, 9, 36, 84, 42, 42, 42, 6, 3, 1
1, 10, 45, 60, 210, 42, 42, 6, 3,1, 1
Let this traingle be called Smarandache AMAR LCM Triangle
Note: As r! divides the product of r consecutive integers so does the LCM ( 1, 2, 3, … r ) divide the LCM of any r consecutive numbers Hence we get only integers as the members of the above triangle.
Following properties of Smarandache AMAR LCM Triangle are noticable.
Some keen observation opens up vistas of challenging problems:
In the 9^{th} row 42 appears at three consecutive places.
OPEN PROBLEM:
(1) Can there be arbitrarily large lengths of equal values appear in a
row.?
SMARANDACHE LCM FUNCTION:
The Smarandache function S(n) is defined as S(n) = k where is the smallest integer such that n divies k! . Here we define another function as follows:
Smarandache Lcm Function denoted by S_{L}( n) = k , where k is
the smallest integer such that n divide LCM ( 1, 2, ,3 . . . k).
Let n = p_{1}^{a1 }p_{2}^{a2} p_{3}^{a3} . . .p_{r}^{ar}
Let p_{m}^{am} be the largest divisor of n with only one prime factor, then
We have S_{L}( n) = p_{m}^{am}
If n =k! then S(n) = k and S_{L}( n)
> k
If n is a prime then we have S_{L}( n) = S(n) = n
Clearly S_{L}( n) ³ S(n) the equality holding good for n a
prime or n = 4 , n=12.
Also S_{L}( n) = n if n is a prime power. (n = p^{a} )
OPEN PROBLEMS:
(1) Are there numbers n >12 for which S_{L}( n) = S(n).
(2) Are there numbers n for which S_{L}( n) = S(n) ¹ n
REFERENCE:
[1] Some new smarandache type sequences, partitions and set. Amarnath
Murthy, SNJ, VOL 1-2-3 , 2000.