OTHER SMARANDACHE TYPE FUNCTIONS
by Jose Castillo
1) Let f: Z ---> Z be a strictly increasing function and x an element
in R. Then:
a) Inferior Smarandache f-Part of x,
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ISf(x) is the smallest k such that f(k) <= x < f(k+1).
b) Superior Smarandache f-Part of x,
--------------------------------
SSf(x) is the smallest k such that f(k) < x <= f(k+1).
Particular cases:
a) Inferior Smarandache Prime Part:
For any positive real number n one defines ISp(n) as the largest
prime number less than or equal to n.
The first values of this function are (Smarandache[6] and Sloane[5]):
2,3,3,5,5,7,7,7,7,11,11,13,13,13,13,17,17,19,19,19,19,23,23.
b) Superior Smarandache Prime Part:
For any positive real number n one defines SSp(n) as the smallest
prime number greater than or equal to n.
The first values of this function are (Smarandache[6] and Sloane[5]):
2,2,2,3,5,5,7,7,11,11,11,11,13,13,17,17,17,17,19,19,23,23,23.
c) Inferior Smarandache Square Part:
For any positive real number n one defines ISs(n) as the largest
square less than or equal to n.
The first values of this function are (Smarandache[6] and Sloane[5]):
0,1,1,1,4,4,4,4,4,9,9,9,9,9,9,9,16,16,16,16,16,16,16,16,16,25,25.
b) Superior Smarandache Square Part:
For any positive real number n one defines SSs(n) as the smallest
square greater than or equal to n.
The first values of this function are (Smarandache[6] and Sloane[5]):
0,1,4,4,4,9,9,9,9,9,16,16,16,16,16,16,16,25,25,25,25,25,25,25,25,25,36.
d) Inferior Smarandache Cubic Part:
For any positive real number n one defines ISc(n) as the largest
cube less than or equal to n.
The first values of this function are (Smarandache[6] and Sloane[5]):
0,1,1,1,1,1,1,1,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,27,27,27,27.
e) Superior Smarandache Cube Part:
For any positive real number n one defines SSs(n) as the smallest
cube greater than or equal to n.
The first values of this function are (Smarandache[6] and Sloane[5]):
0,1,8,8,8,8,8,8,8,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27,27.
f) Inferior Smarandache Factorial Part:
For any positive real number n one defines ISf(n) as the largest
factorial less than or equal to n.
The first values of this function are (Smarandache[6] and Sloane[5]):
1,2,2,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,24,24,24,24,24,24,24.
g) Superior Smarandache Factorial Part:
For any positive real number n one defines SSf(n) as the smallest
factorial greater than or equal to n.
The first values of this function are (Smarandache[6] and Sloane[5]):
1,2,6,6,6,6,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,120.
Remark 1: This is a generalization of the inferior/superior integer
part of a number.
2) Let f: Z ---> Z be a strictly increasing function and x an element
in R. Then:
Fractional Smarandache f-Part of x,
----------------------------------
FSf(x) = x - ISf(x),
where ISf(x) is the Inferior Smarandache f-Part of x defined above.
Particular cases:
a) Fractional Smarandache Prime Part:
FSp(x) = x - ISp(x),
where ISp(x) is the Inferior Smarandache Prime Part defined above.
Example: FSp(12.501) = 12.501 - 11 = 1.501.
b) Fractional Smarandache Square Part:
FSs(x) = x - ISs(x),
where ISs(x) is the Inferior Smarandache Square Part defined above.
Example: FSs(12.501) = 12.501 - 9 = 3.501.
c) Fractional Smarandache Cubic Part:
FSc(x) = x - ISc(x),
where ISc(x) is the Inferior Smarandache Cubic Part defined above.
Example: FSc(12.501) = 12.501 - 8 = 4.501.
d) Fractional Smarandache Factorial Part:
FSf(x) = x - ISf(x),
where ISf(x) is the Inferior Smarandache Factorial Part defined above.
Example: FSf(12.501) = 12.501 - 6 = 6.501.
Remark 2.1: This is a generalization of the fractional part of a number.
Remark 2.2: In a similar way one defines:
- the Inferior Fractional Smarandache f-Part:
IFSf(x) = x - ISf(x) = FSf(x);
- and the Superior Fractional Smarandache f-Part:
SFSf(x) = SSf(x) - x;
for example: Superior Fractional Smarandache Cubic Part of 12.501
= 27 - 12.501 = 14.499.
3) Let g: A ---> A be a strictly increasing function, and let "~" be a
given internal law on A. Then we say that
f: A ---> A is smarandachely complementary with respect to the
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function g and the internal law "~" if:
-----------------------------------
f(x) is the smallest k such that there exists a z in A so that
x~k = g(z).
Particular cases:
a) Smarandache Square Complementary Function:
f: N ---> N, f(x) = the smallest k such that xk is a
perfect square.
The first values of this function are (Smarandache[6] and Sloane[5]):
1,2,3,1,5,6,7,2,1,10,11,3,14,15,1,17,2,19,5,21,22,23,6,1,26,3,7.
b) Smarandache Cubic Complementary Function:
f: N ---> N, f(x) = the smallest k such that xk is a
perfect cube.
The first values of this function are (Smarandache[6] and Sloane[5]):
1,4,9,2,25,36,49,1,3,100,121,18,169,196,225,4,289,12,361,50.
More generally:
c) Smarandache m-power Complementary Function:
f: N ---> N, f(x) = the smallest k such that xk is a
perfect m-power.
d) Smarandache Prime Complementary Function:
f: N ---> N, f(x) = the smallest k such that x+k is a prime.
The first values of this function are (Smarandache[6] and Sloane[5]):
1,0,0,1,0,1,0,3,2,1,0,1,0,3,2,1,0,1,0,3,2,1,0,5,4,3,2,1,0,1,0,5.
4) Smarandache-Multiplicative Function:
* *
A function f : N --> N which,
for any (a, b) = 1, f(ab) = max {f(a), f(b)};
[i.e. it reflects the main property of the Smarandache function].
Reference:
[1] Tabirca, Sabin, "About S-Multiplicative Functions", ,
Brasov, Vol. 7, No. 1, 169-170, 1999.
References:
[1] Castillo, Jose, "Other Smarandache Type Functions",
http://www.gallup.unm.edu/~smarandache/funct2.txt
[2] Dumitrescu, C., Seleacu, V., "Some Notions and Questions in
Number THeory", Xiquan Publ. Hse., Phoenix-Chicago, 1994.
[3] Popescu, Marcela, Nicolescu, Mariana, "About the Smarandache
Complementary Cubic Function", ,
Vol. 7, no. 1-2-3, 54-62, 1996.
[4] Popescu, Marcela, Seleacu, Vasile, "About the Smarandache
Complementary Prime Function", ,
Vol. 7, no. 1-2-3, 12-22, 1996.
[5] Sloane, N.J.A.S, Plouffe, S., "The Encyclopedia of Integer
Sequences", online, email: superseeker@research.att.com
(SUPERSEEKER by N. J. A. Sloane, S. Plouffe, B. Salvy, ATT
Bell Labs, Murray Hill, NJ 07974, USA).
[6] Smarandache, Florentin, "Only Problems, not Solutions!", Xiquan
Publishing House, Phoenix-Chicago, 1990, 1991, 1993;
ISBN: 1-879585-00-6.
(reviewed in by P. Kiss: 11002,
744, 1992;
and in , Aug.-Sept. 1991);
[7] "The Florentin Smarandache papers" Special Collection, Arizona State
University, Hayden Library, Tempe, Box 871006, AZ 85287-1006, USA;
(Carol Moore & Marilyn Wurzburger: librarians).