Constant

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 <Smarandache Notions Journal>, 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 <Smarandache Notions Journal>, 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 <Smarandache Notions Journal>, 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 <Smarandache Notions Journal>, University of Craiova, Vol. 7, No. 1-2-3, pp. 121-126, August 1996. 5)

The series Sigma (-1) n>=1

n-1 (S(n)/n!)

converges to an irrational number. Reference: [1] Sandor, Jozsef, "On The Irrationality Of Certain Alternative Smarandache Series", <Smarandache Notions Journal>, 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", <Smarandache Function Journal>, 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", <Smarandache Notions Journal>, 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", <Smarandache Notions Journal>, 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", <Smarandache Notions Journal>, 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", <Smarandache Notions Journal>, 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", <Smarandache Notions Journal>, 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", <Smarandache Notions Journal>, 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", <Smarandache Notions Journal>, Vol. 7, No. 1-2-3, August 1996, pp. 7-9.