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.