**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 r^{th }even number.

We have E(1) = 2, E(2) = 4, or E^{2} ( 1) =4, E(4) = 8 or E^{3}
(1) = 8 etc.

We have E^{n} (1) = 2^{n}

**[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 O^{n} (2) = 2^{n} + 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, . . .

T_{n} = T_{n-1}^{th }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

T_{1} = 3, and T_{n-1} = d( T_{n}) , the number of
divisors of T_{n} , where T_{n} is smallest such number.

3, 4, 6, 12, 72, 2^{8}.3^{7}, 2^{2186} x 3^{255}
, **. . . { **where 3^{7}-1 = 2186 and 2^{8}-1 = 255 }

3, 4, 6, 12, 72, 559872, 2^{2186} x 3^{255} , **. . .**

The sequence obtained by incrementing the above sequence by 1 is

4, 5, 7, 13, 73, 559873, 2^{2186} x 3^{255} + 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,
**. . .** - 2, 3, 4, 7, 8, 15, 23,
**24**,52,**. . .** - 5, 6, 12,
**28**,**56**, 120, 240, 744, 1920,**. . .** - 9, 13, 14,
**24**,**. . .** - 10,
**18,**39,**56,****. . .** - 11,
**12,****28**,**. . .** - 16, 31, 32, 63, 104,
**. . .** - 17
**, 18, . . .** - 19, 20, 42
**, . . .**

In the above sequences T_{n }= s
(T_{n-1}) , with T_{1} 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, . . .
- 2, 1, . . .
- 3, 1, . . .
- 4, 3, 1, . . .
- 5, 1,. . .
**6, 6, 6, . . .****7,**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?

** **