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 [quote="CrystyB"]mith, have you read this topic anytime during the last year? (IOW bump)[/quote]
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CrystyB
Posted: Wed Oct 31, 2001 5:38 pm    Post subject: 1

mith, have you read this topic anytime during the last year?

(IOW bump)
CrystyB
Posted: Tue Aug 21, 2001 8:25 am    Post subject: 0

mith, i wonder how do you pronounce the last five letter of his name: dash, or da-ke (as in DAkota, KEntucky)?

Second, i have a few stuff to ask if you don't mind:
SK(3) is said to be 4, [but !4=10] and i think there is no such number. I think i can prove that too.
i don't get the meaning of Sk(n). The examples didn't help me either.
there is no SNTP(4x) for any integer x. I think i can prove that too.
is it easy to show that Z(F(p))=p?
where did you found the problem of Z(n)=Z(n+1) instead of S(...)?

TYVMia,
Cristian
CrystyB
Posted: Mon Aug 20, 2001 7:03 pm    Post subject: -1

code:

FUNCTIONS IN NUMBER THEORY

1) Smarandache-Kurepa Function:
For p prime, SK(p) is the smallest integer such that !SK(p) is
divisible by p, where !SK(p) = 0! + 1! + 2! + ... + (p-1)!

For example:
p 2 3 7 11 17 19 23 31 37 41 61 71 73 89
SK(p) 2 4 6 6 5 7 7 12 22 16 55 54 42 24

2) Smarandache-Wagstaff Function:
For p prime, SW(p) is the smallest integer such that W(SW(p)) is
divisible by p, where W(p) = 1! + 2! + ... + (p)!

For example:
p 3 11 17 23 29 37 41 43 53 67 73 79 97
SW(p) 2 4 5 12 19 24 32 19 20 20 7 57 6

3) Smarandache Ceil Functions of n-th Order:
Sk(n) is the smallest integer for which n divides Sk(n)^k.

For example, for k=2, we have:
n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
S2(n) 2 4 3 6 10 12 5 9 14 8 6 20 22 15 12 7

4) Pseudo-Smarandache Function:
Z(n) is the smallest integer such that 1 + 2 + ... + Z(n) is divisible
by n.

For example:
n 1 2 3 4 5 6 7
Z(n) 1 3 2 3 4 3 6

5) Smarandache Near-To-Primordial Function:
* * *
SNTP(n) is the smalest prime such that either p - 1, p , or p + 1
is divisible by n,
*
where p , of a prime number p, is the product of all primes less than
or equal to p.

For example:

n 1 2 3 4 5 6 7 8 9 10 11 ... 59 ...

SNTP(n) 2 2 2 5 3 3 3 5 ? 5 11 ... 13 ...

6) Smarandache Double-Factorial Function:
SDF(n) is the smallest number such that SDF(n)!! is divisible by n,
where the double factorial
m!! = 1x3x5x...xm, if m is odd;
and m!! = 2x4x6x...xm, if m is even.

For example:

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

SDF(n) 1 2 3 4 5 6 7 4 9 10 11 6 13 14 5 6

7) Smarandache Primitive Functions:

Let p be a positive prime.
n
S : N ---> N, having the property that (S (n))! is divisible by p ,
p p

and it is the smallest integer with this property.

For example:

S (4) = 9, because 9! is divisible by 3^4, and it is the smallest one
3
with this property.

These functions help computing the Smarandache Function.

8) Smarandache Function:

S : N ---> N, S(n) is the smallest integer such that S(n)! is
divisible by n.

9) Smarandache Functions of the First Kind:

* *
S : N --> N
n
r
i) If n = u (with u = 1, or u = p prime number), then

S (a) = k, where k is the smallest positive integer such that
n
ra
k! is a multiple of u ;

r1 r2 rt
ii) If n = p1 . p2 ... pt , then S (a) = max { S (a) }.
n 1<=j<=t rj
pj

10) Smarandache Functions of the Second Kind:

k * * k *
S : N --> N , S (n) = S (k) for k in N ,
n

where S are the Smarandache functions of the first kind.
n

11) Smarandache Function of the Third Kind:

b
S (n) = S (b ), where S is the Smarandache function of the
a a n a
n n

first kind, and the sequences (a ) and (b ) are different from
n n

the following situations:
*
i) a = 1 and b = n, for n in N ;
n n
*
ii) a = n and b = 1, for n in N .
n n

Hope this helps as a reference (it took me a long time to find this text through all those links...)