“Smarandache Function”的概念、定义、习题解题方法及翻译-数学辞典

单词

Smarandache Function

释义

Smarandache Function

The smallest value for a given for which ( divides Factorial). For example, the number8 does not divide , , , but does divide , so . For a Prime ,, and for an Even Perfect Number , is Prime (Ashbacher 1997).

The Smarandache numbers for , 2, ... are 1, 2, 3, 4, 5, 3, 7, 4, 6, 5, 11, ... (Sloane's A002034). Letting denote the smallest value of for which , 2, ..., then is given by 1, 2, 3, 4, 5, 9, 7, 32, 27, 25, 11,243, ... (Sloane's A046021). Some values of first occur only for very large , for example, ,, , , and . D. Wilson points out that if we let

be the power of the Prime

, where

is the sum of the base-

digits of

, then it followsthat

where the minimum is taken over the Primes

dividing

. This minimum appears to always be achieved when

is theGreatest Prime Factor of

The incrementally largest values of are 1, 2, 3, 4, 5, 7, 11, 13, 17, 19, 23, 29, ... (Sloane's A046022), which occurfor , 2, 3, 4, 5, 7, 11, 13, 17, 19, 23, 29, ... (Sloane's A046023), i.e., the values where .

Tutescu (1996) conjectures that the Diophantine Equation has no solution.

See also Factorial, Greatest Prime Factor, Pseudosmarandache Function, Smarandache Ceil Function,Smarandache Constants, Smarandache-Kurepa Function, Smarandache Near-to-Primorial Function,Smarandache-Wagstaff Function

References

Ashbacher, C. An Introduction to the Smarandache Function. Cedar Rapids, IA: Decisionmark, 1995.

Ashbacher, C. ``Problem 4616.'' School Sci. Math. 97, 221, 1997.

Begay, A. ``Smarandache Ceil Functions.'' Bulletin Pure Appl. Sci. India 16E, 227-229, 1997.

Dumitrescu, C. and Seleacu, V. The Smarandache Function. Vail, AZ: Erhus University Press, 1996.

``Functions in Number Theory.'' http://www.gallup.unm.edu/~smarandache/FUNCT1.TXT.

Ibstedt, H. Surfing on the Ocean of Numbers--A Few Smarandache Notions and Similar Topics. Lupton, AZ: Erhus University Press, pp. 27-30, 1997.

Sandor, J. ``On Certain Inequalities Involving the Smarandache Function.'' Abstracts of Papers Presented to the Amer. Math. Soc. 17, 583, 1996.

Sloane, N. J. A. Sequences A046021,A046022,A046023, andA002034/M0453in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html.

Smarandache, F. Collected Papers, Vol. 1. Bucharest, Romania: Tempus, 1996.

Smarandache, F. Collected Papers, Vol. 2. Kishinev, Moldova: Kishinev University Press, 1997.

Tutescu, L. ``On a Conjecture Concerning the Smarandache Function.'' Abstracts of Papers Presented to the Amer. Math. Soc. 17, 583, 1996.

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