The Smarandache ceil function 
is defined for positive
integers 
and 
as the smallest positive integer for which ![n|[S_k(n)]^k](/images/equations/SmarandacheCeilFunction/Inline4.svg)
.
For a prime factorization 
, it is given by ![S_k(n)=product_(i)p_i^([e_i/k])](/images/equations/SmarandacheCeilFunction/Inline6.svg)
, where ![[x]](/images/equations/SmarandacheCeilFunction/Inline7.svg)
is the ceiling function.
It also satisfies
where 
is the number of residue classes 
modulo 
satisfying 
.
The functions 
for 
, 3, ..., 6 for values such that 
are tabulated by Begay (1997). The following table
gives 
for small 
and 
,
2, ....
 | OEIS |  |
| 1 | A000027 | 1, 2, 3, 4, 5, 6,
7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ... |
| 2 | A019554 | 1,
2, 3, 2, 5, 6, 7, 4, 3, 10, 11, 6, 13, 14, 15, 4, 17, 6, ... |
| 3 | A019555 | 1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 4, 17, 6, ... |
| 4 | A053166 | 1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, ... |
See also
Pseudosmarandache Function,
Smarandache Function,
Smarandache-Kurepa
Function,
Smarandache Near-to-Primorial
Function,
Smarandache Sequences,
Smarandache-Wagstaff Function
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References
Begay, A. "Smarandache Ceil Functions." Bull. Pure Appl. Sci. 16E, 227-229, 1997. https://fs.unm.edu/SMARCEIL.HTM.Sloane,
N. J. A. Sequences A000027/M0472,
A019554, A019555,
and A053166 in "The On-Line Encyclopedia
of Integer Sequences."Smarandache, F. Only
Problems, Not Solutions!, 4th ed. Phoenix, AZ: Xiquan, 1993.Smarandache,
F. Collected Papers, Vol. 2. Kishinev, Moldova: Kishinev University Press,
1997.Weisstein, E. W. "Two Gigantic Primes with Prime Digits
Found." MathWorld Headline News, Apr. 9, 2002. https://mathworld.wolfram.com/news/2002-04-09/primeprimes/.Referenced
on Wolfram|Alpha
Smarandache Ceil Function
Cite this as:
Weisstein, Eric W. "Smarandache Ceil Function."
From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SmarandacheCeilFunction.html
Subject classifications
