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 Tn = 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 rth kind as SLRS(r)

 

The n th term rTn =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, . . . 1Tn (= n.)

we get SLRS(2) as

1, 3, 6, 10, . . . 2Tn = 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 rTn .

 

Definition: Like nCr , the combination of r out of n given objects , We define a new term nLr

As

nLr = 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 kth column is nothing but the SLRS(k).
  3. The first four rows are the same as that of the Pascal's Triangle.
  4. IInd column contains natural numbers.
  5. IIIrd column elements are the triangular numbers.
  6. If p is a prime then p divides all the terms of the pth row except the first and the last which are unity. In other words S pth row 2 ( mod p)

 

Some keen observation opens up vistas of challenging problems:

In the 9th row 42 appears at three consecutive places.

OPEN PROBLEM:

(1) Can there be arbitrarily large lengths of equal values appear in a row.?

  1. To find the sum of a row.
  2. 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 SL( n) = k , where k is the smallest integer such that n divide LCM ( 1, 2, ,3 . . . k).

Let n = p1a1 p2a2 p3a3 . . .prar

Let pmam be the largest divisor of n with only one prime factor, then

We have SL( n) = pmam

If n =k! then S(n) = k and SL( n) > k

If n is a prime then we have SL( n) = S(n) = n

Clearly SL( n) S(n) the equality holding good for n a prime or n = 4 , n=12.

Also SL( n) = n if n is a prime power. (n = pa )

OPEN PROBLEMS:

(1) Are there numbers n >12 for which SL( n) = S(n).

(2) Are there numbers n for which SL( n) = S(n) n

 REFERENCE:

[1] Some new smarandache type sequences, partitions and set. Amarnath Murthy, SNJ, VOL 1-2-3 , 2000.