** In this paper some definitions, examples and conjectures are
exposed related to the Smarandache type functions, found in the Archives
of the Arizona State University, Tempe, USA Special Collections.**

**(1) Smarandache Ceil Function of Second Order:
2, 4, 3, 6, 4, 6, 10, 12, 5, 9, 14, 8, 6, 20, 22, 15, 12, 7, 10, 26, 18,
28, 30, 21, 8, 34, 12, 15, 38, 20, 9, 42, 44, 30, 46, 24, 14, 33, 10, 52,
18, 28, 58, 39, 60, 11, 62, 25, 42, 16, 66, 45, 68, 70, 12, 21, 74, 30,
76, 51, 78, 40, 18, 82, 84, 13, 57, 86, ...
(S**

**(2) Smarandache Ceil Function of Third Order:
2, 2, 3, 6, 4, 6, 10, 6, 5, 3, 14, 4, 6, 10, 22, 15, 12, 7, 10, 26, 6,
14, 30, 21, 4, 34, 6, 15, 38, 20, 9, 42, 22, 30, 46, 12, 14, 33, 10, 26,
6, 28, 58, 39, 30, 11, 62, 5, 42, 8, 66, 15, 34, 70, 12, 21, 74, 30, 38,
51, 78, 20, 18, 82, 42, 13, 57, 86, ...
(S**

**(3) Smarandache Ceil Function of Fourth Order:
2, 2, 3, 6, 2, 6, 10, 6, 5, 3, 14, 4, 6, 10, 22, 15, 6, 7, 10, 26, 6, 14,
30, 21, 4, 34, 6, 15, 38, 10, 3, 42, 22, 30, 46, 12, 14, 33, 10, 26, 6,
14, 58, 39, 30, 11, 62, 5, 42, 4, 66, 15, 34, 70, 6, 21, 74, 30, 38, 51,
78, 20, 6, 82, 42, 13, 57, 86, ...
(S**

**(4) Smarandache Ceil Function of Fifth Order:
2, 2, 3, 6, 2, 6, 10, 6, 5, 3, 14, 2, 6, 10, 22, 15, 6, 7, 10, 26, 6, 14,
30, 21, 4, 34, 6, 15, 38, 10, 3, 42, 22, 30, 46, 6, 14, 33, 10, 26, 6,
14, 58, 39, 30, 11, 62, 5, 42, 4, 66, 15, 34, 70, 6, 21, 74, 30, 38, 51,
78, 10, 6, 82, 42, 13, 57, 86, ...
(S**

**(5) Smarandache Ceil Function of Sixth Order:
2, 2, 3, 6, 2, 6, 10, 6, 5, 3, 14, 2, 6, 10, 22, 15, 6, 7, 10, 26, 6, 14,
30, 21, 2, 34, 6, 15, 38, 10, 3, 42, 22, 30, 46, 6, 14, 33, 10, 26, 6,
14, 58, 39, 30, 11, 62, 5, 42, 4, 66, 15, 34, 70, 6, 21, 74, 30, 38, 51,
78, 10, 6, 82, 42, 13, 57, 86, ...
(S**

**(6) Smarandache - Fibonacci triplets:
11, 121, 4902, 26245, 32112, 64010, 368140, 415664, 2091206, 2519648, 4573053,
7783364, 79269727, 136193976, 321022289, 445810543, 559199345, 670994143,
836250239, 893950202, 937203749, 1041478032, 1148788154, ...
(An integer n such that S(n) = S(n-1) + S(n-2) where S(k) is the Smarandache
function: the smallest number k such that S(k)! is divisible by k.)
Remarks:
It is not known if this sequence has infinitely or finitely many terms.
H. Ibstedt and C. Ashbacher independently conjectured that there are infinitely
many.
H. I. found the biggest known number: 19 448 047 080 036.
References:
(a) Surfing on the Ocean of Numbers -- a few Smarandache Notions and Similar
Topics, by Henry Ibstedt, Erhus University Press, Vail, USA, 1997; p. 19-23.
**

**(b) C. Ashbacher and M. Mudge, , London,
October 1995; p. 302. **

**(7) Smarandache-Radu duplets
224, 2057, 265225, 843637, 6530355, 24652435, 35558770, 40201975, 45388758,
46297822, 67697937, 138852445, 157906534, 171531580, 299441785, 551787925,
1223918824, 1276553470, 1655870629, 1853717287, 1994004499, 2256222280,
...
(An integer n such that between S(n) and S(n+1) there is no prime [S(n)
and S(n + 1) included].
where S(k) is the Smarandache function: the smallest number k such that
S(k)! is divisible by k.)
Remarks:
It is not known if this sequence has infinitely or finitely many terms.
H. Ibstedt conjectured that there are infinitely many.
H. I. found the biggest known number:
270 329 975 921 205 253 634 707 051 822 848 570 391 313!**

**References:
(a) Surfing on the Ocean of Numbers -- a few Smarandache Notions and Similar
Topics, by Henry Ibstedt, Erhus University Press, Vail, USA, 1997; p. 19-23.
(b) I. M. Radu, Mathematical Spectrum, Sheffield University, UK, Vol. 27,
(No. 2), 1994/5; p. 43.**

*** Originally appeared in Bulletin of Pure and Applied Sciences, Vol.
16E(No. 2), 1997; p. 227-229. **