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
Reference:
[1] C.Ashbacher, "Some Properties of the Smarandache-Kurepa and
Smarandache-Wagstaff Functions", in , Vol. 7, No. 3, pp. 114-116, September 1997.
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
Reference:
[1] C.Ashbacher, "Some Properties of the Smarandache-Kurepa and
Smarandache-Wagstaff Functions", in , Vol. 7, No. 3, pp. 114-116, September 1997.
3) Smarandache Ceil Functions of k-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
References:
[1] H.Ibstedt, "Surfing on the Ocean of Numbers -- A Few Smarandache
Notions and Similar Topics", Erhus Univ. Press, Vail, USA, 1997;
pp. 27-30.
[2] A.Begay, "Smarandache Ceil Functions", in , India, Vol. 16E, No. 2, 1997, pp. 227-229.
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
Reference:
[1] K.Kashihara, "Comments and Topics on Smarandache Notions and
Problems", Erhus University Press, Vail, USA, 1996.
5) Smarandache Near-To-Primordial Function:
* * *
SNTP(n) is the smallest 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 ...
References:
[1] Mudge, Mike, "The Smarandache Near-To-Primordial (S.N.T.P.) Function",
, Vol. 7, No. 1-2-3, August 1996, p. 45.
[2] Ashbacher, Charles, "A Note on the Smarandache Near-To-Primordial
Function", , Vol. 7, No. 1-2-3, August 1996,
pp. 46-49.
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
Reference:
[1[] Dumitrescu, C., Seleacu, V., "Some notions and questions in number
theory", Erhus Univ. Press, Glendale, 1994, Section #54 ("Smarandache
Double Factorial Numbers").
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.
Reference:
[1] Smarandache, Florentin, "A function in number theory", , Seria St. Mat., Vol. XVIII, fasc. 1, 1980,
pp. 79-88.
8) Smarandache Function:
S : N ---> N, S(n) is the smallest integer such that S(n)! is
divisible by n.
Reference:
[1] Smarandache, Florentin, "A function in number theory", , Seria St. Mat., Vol. XVIII, fasc. 1, 1980,
pp. 79-88.
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
Reference:
[1] Balacenoiu, Ion, "Smarandache Numerical Functions", , Vol. 14E, No. 2, 1995, pp. 95-100.