SMARANDACHE FUNCTION OF A FUNCTION AND OTHER SEQUENCES

(Amarnath Murthy, S.E.(E&T) , WLS, Oil and Natural Gas Corporation Ltd., Sabarmati, Ahmedabad,- 380005 INDIA. )

 Consider the even function E(r) = the rth even number.

We have E(1) = 2, E(2) = 4, or E2 ( 1) =4, E(4) = 8 or E3 (1) = 8 etc.

We have En (1) = 2n

[1] SMARANDACHE EVEN-EVEN SEQUENCE is defined as

1, 2 , 4, 8, 16,. . .

Similarly

[2] SMARANDACHE ODD-ODD SEQUENCE is defined as

O(2) =3 , O(3) = 5 , O(5) = 9 etc. {as O(1) =1 we start with O(2)}

2, 3, 5, 9, 17, 33. . . in which On (2) = 2n + 1

[3] SMARANDACHE PRIME-PRIME SEQUENCE is defined as

P(1) = 2, P(2) = 3, P(3) = 5, P(5) = 11, P(11) = 31,

2, 3, 5, 11, 31, . . .

Tn = Tn-1th Prime.

[4] SMARANDACHE TRAINGULAR -TRIANGULAR NUMBER SEQUENCE is defined as

T(2) = 3, T(3) = 6, T(6) = 21,T(21) = 231, T(231) = 26796

2, 3, 6, 21, 231, 26796, . . .

We can generate innumerable number of sequences like this.

[5] SMARANDACHE DIVISORS OF DIVISORS SEQUENCE is defined as follows

T1 = 3, and Tn-1 = d( Tn) , the number of divisors of Tn , where Tn is smallest such number.

3, 4, 6, 12, 72, 28.37, 22186 x 3255 , . . . { where 37-1 = 2186 and 28-1 = 255 }

3, 4, 6, 12, 72, 559872, 22186 x 3255 , . . .

  The sequence obtained by incrementing the above sequence by 1 is

4, 5, 7, 13, 73, 559873, 22186 x 3255 + 1 , . . .

CONJECTURE: The above sequence contains all primes from the second term onwards.

The motivation behind this conjecture : As the neighboring number is highly composite ( the smallest number having such a given large number of divisors), the chances of it being a prime is very high.

[6]SMARANDACHE DIVISOR SUM-DIVISOR SUM SEQUENCES (SDSDS) are defined as follows: Consider the following sequences in witch each term is the sum of the divisors of the previous term:

  1. 1, 1, 1, 1, 1, 1, . . .
  2. 2, 3, 4, 7, 8, 15, 23, 24 ,52, . . .
  3. 5, 6, 12, 28, 56, 120, 240, 744, 1920, . . .
  4. 9, 13, 14, 24, . . .
  5. 10, 18, 39, 56, . . .
  6. 11, 12, 28, . . .
  7. 16, 31, 32, 63, 104, . . .
  8. 17, 18, . . .
  9. 19, 20, 42, . . .

In the above sequences Tn = s (Tn-1) , with T1 as the generator of the sequence. A number which has occurred in a previous sequence is not to be taken as a generator.

Problems: (1) How many of the numbers like 12, 18, 24, 28, 56 etc. are members of two or more sequences ?

(2) Are there numbers which are members of more than two sequences?

We define the SMARANDACHE DIVISOR SUM GENERATOR SEQUENCE (SDSGS) as the sequence formed by ( the generators) the first terms of each of the above sequences.

1, 2, 5, 9, 10, 11, 16, 17, 19, . . .

PROBLEM: Is the above sequence finite or infinite?

SMARANDACHE REDUCED DIVISOR SUM PERIODICITY SEQUENCES:

In the following sequences the sum of the proper divisors only is taken till the sequence terminates at 'one ' or repeats itself.

  1. 1, 1, 1, . . .
  2. 2, 1, . . .
  3. 3, 1, . . .
  4. 4, 3, 1, . . .
  5. 5, 1,. . .
  6. 6, 6, 6, . . .
  7. 7, 1, . . .
  1. 12, 16, 15, 8, 7, 1 , etc.

220, 284 , 220, 284, . . .

We define the life of a number by the number of terms in the corresponding sequence till a 'one' is arrived at. e.g. the life of 2 is 2 that of 12 is 6 etc. The life of a perfect number like 6 or 28 etc. or that of a amicable number pair like (220, 284) is infinite. The same is true for a sociable number like the five number chain 12496, 14288, 15472, 14536, 14264. We can call them immortals.

PROBLEM: Given an arbitrary number n . Is there any number whose life is n?