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, -------------------------------- 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 ----------------------------------------------- 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).