CONSTANTS INVOLVING THE SMARANDACHE FUNCTION Let S(n) be the Smarandache Function, i.e. the smallest integer such that S(n)! is divisible by n. 1) The First Constant of Smarandache: Sigma(1/S(n)!) is convergent to a number s1 between 0.000 and 0.717. n>=2 Reference: [1] I.Cojocaru, S.Cojocaru, "The First Constant of Smarandache", in , University of Craiova, Vol. 7, No. 1-2-3, pp. 116-118, August 1996. 2) The Second Constant of Smarandache: Sigma(S(n)/n!) is convergent to an irrational number s2. n>=2 Reference: [1] I.Cojocaru, S.Cojocaru, "The Second Constant of Smarandache", in , University of Craiova, Vol. 7, No. 1-2-3, pp. 119-120, August 1996. 3) The Third Constant of Smarandache: Sigma(1/(S(2)S(3)...S(n))) is convergent to a number s3, which is n>=2 between 0.71 and 1.01. Reference: [1] I.Cojocaru, S.Cojocaru, "The Third and Fourth Constants of Smarandache", in , University of Craiova, Vol. 7, No. 1-2-3, pp. 121-126, August 1996. 4) The Fourth Constant of Smarandache: Sigma(n^alpha/(S(2)S(3)...S(n))), where alpha >= 1, n>=2 is convergent to a number s4. Reference: [1] I.Cojocaru, S.Cojocaru, "The Third and Fourth Constants of Smarandache", in , University of Craiova, Vol. 7, No. 1-2-3, pp. 121-126, August 1996. 5) The series n-1 Sigma (-1) (S(n)/n!) n>=1 converges to an irrational number. Reference: [1] Sandor, Jozsef, "On The Irrationality Of Certain Alternative Smarandache Series", , Vol. 8, No. 1-2-3, Fall 1997, pp. 143-144. 6) The series S(n) Sigma -------- n>=2 (n+1)! converges to a number s6, where e-3/2 < s6 < 1/2. Reference: [1] Burton, Emil, "On Some Series Involving the Smarandache Function", , Vol. 6, No. 1, June 1995, ISSN 1053-4792, pp. 13-15. [2] Dumitrescu, C., Seleacu, V., "The Smarandache Function", Erhus University Press, Vail, USA, 1996, pp. 48-61 (chapter "Numerical Series Containing the Function S"). 7) The series S(n) Sigma --------, where r is a natural number, n>=r (n+r)! converges to a number s7. Reference: [1] Dumitrescu, C., Seleacu, V., "The Smarandache Function", Erhus University Press, Vail, USA, 1996, pp. 48-61 (chapter "Numerical Series Containing the Function S"). 8) The series S(n) Sigma --------, where r is a nonzero natural number, n>=r (n-r)! converges to a number s8. Reference: [1] Dumitrescu, C., Seleacu, V., "The Smarandache Function", Erhus University Press, Vail, USA, 1996, pp. 48-61 (chapter "Numerical Series Containing the Function S"). 9) The series 1 Sigma -------------------- n>=2 n Sigma (S(i)!/i) i=2 is convergent to a number s9. Reference: [1] Dumitrescu, C., Seleacu, V., "The Smarandache Function", Erhus University Press, Vail, USA, 1996, pp. 48-61 (chapter "Numerical Series Containing the Function S"). 10) The series 1 Sigma -------------------, where alpha > 1, n>=2 alpha ______ S(n) \/S(n)! is convergent to a number s10. References: [1] Burton, Emil, "On Some Convergent Series", , Vol. 7, No. 1-2-3, August 1996, pp. 7-9. [2] Dumitrescu, C., Seleacu, V., "The Smarandache Function", Erhus University Press, Vail, USA, 1996, pp. 48-61 (chapter "Numerical Series Containing the Function S"). 11) The series 1 Sigma -----------------------, where alpha > 1, n>=2 alpha _________ S(n) \/(S(n)-1)! is convergent to a number s11. References: [1] Burton, Emil, "On Some Convergent Series", , Vol. 7, No. 1-2-3, August 1996, pp. 7-9. * 12) Let f : N ----> R be a function which satisfies the condition c f(t) <= ------------------------------- alpha t (d(t!)) - d((t-1)!) for t a nonzero natural number, d(x) the number of divisors of x, and the given constants alpha > 1, c > 1. Then the series Sigma f(S(n)) n>=1 is convergent to a number s11 . f Reference: [1] Burton, Emil, "On Some Convergent Series", , Vol. 7, No. 1-2-3, August 1996, pp. 7-9. 13) The series 1 Sigma ------------------ n>=1 n n ( Product S(k)! ) k=2 is convergent to a number s13. Reference: [1] Burton, Emil, "On Some Convergent Series", , Vol. 7, No. 1-2-3, August 1996, pp. 7-9. 14) The series 1 Sigma -----------------------------, where p > 1, n>=1 _____ p S(n)! \/S(n)! (log S(n)) is convergent to a number s14. Reference: [1] Burton, Emil, "On Some Convergent Series", , Vol. 7, No. 1-2-3, August 1996, pp. 7-9. 15) The series n 2 Sigma -------------, n>=1 n S(2 )! is convergent to a number s15. Reference: [1] Burton, Emil, "On Some Convergent Series", , Vol. 7, No. 1-2-3, August 1996, pp. 7-9. 16) The series S(n) Sigma --------, where p is a real number > 1, n>=1 1+p n converges to a number s16. (For 0 <= p <= 2 the series diverges.) Reference: [1] Burton, Emil, "On Some Convergent Series", , Vol. 7, No. 1-2-3, August 1996, pp. 7-9.