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.