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 _rT_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, . . . ₁T_n (= n.)

we get SLRS(2) as

1, 3, 6, 10, . . . ₂T_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 _rT_n .

Definition: Like ⁿC_{r ,} the combination of r out of n given objects , We define a new term ⁿL_r

As

ⁿL_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 ¹L₀ =1, ¹L₁ =1, ²L₀ =1, ²L₁ =2, ²L₂ =2 etc. define ⁰L₀ =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.

1. The first column and the leading diagonal elements are all unity.
2. The k^th column is nothing but the SLRS(k).
3. The first four rows are the same as that of the Pascal's Triangle.
4. II^nd column contains natural numbers.
5. III^rd column elements are the triangular numbers.
6. If p is a prime then p divides all the terms of the p^th row except the first and the last which are unity. In other words S p^th row º 2 ( mod p)

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.?

2. To find the sum of a row.
3. Explore for congruence properties for composite n.

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 = p₁^a1 p₂^a2 p₃^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.