A return visit to the Florentin Smarandache Function

1 Pages·1993·0.1 MB·

by  Mike Mudge

Mike Mudge pays a n Review of '~umbers Count -118-
return visit to the Florentin r-iiri't It may be advantageous to con February 1993: a revisit to T:,e
sider the STA""TIARD FOR.\f of n, viz n Florentin Smarandache Function 2
Smarandache Function.l = ep~: p;, p~ ...... p~, where e=::l, and This produced a number of 'quite
P"P,.P]' .. p, denote the distinct prime powerful' responses. As a note of
factors of n and a" a" a, ... a, are their related interest. the latest publication
The originator of this function. respective multiplicities. of FLSmarandache is 'A ~umerical
Florentin Smarandache. an Eastern NOTE Sin) is an even function, bv Function in Congruence Theory',
European mathematician. escaped from which is meant S(-n) = S(n). . Libertas .Wathematica (American
the countrY of his birth because the Problem (i) Using either graphical or Romanian Academv of Arts and
Communist authorities had orohibited finite difference technique (Le. the con Science) vol 12, 1992. DO 181-185.
the publication of his rese~ch papers struction of difference tables etc) or .A.riington. Texas. • ,
and his participation in international indeed anything else that comes to Pal Gronas of Norwa\, submitted
congresses. After spending two years mind. address the following questions: theoretical results on both problems
in a political refugee camp in Turkey. (a) Is there a closed expression (for o & (v). However. the clear winner
he emigrated to the United States. mula) for Sin)? this month is a former regular respon
Robert~fullerofThe Number Theorv (b) Is there a good asymptotic expres dent. now retired. Henr ... Ibstedt,
Publishing Company, PO Box 42561, sion for Sin)? (Bv which is meant a Glimminge 2036. 280 60 Brobv,
Phoenix. Arizona 85080, USA, decided formula. which although never (in gen Sweden. Henry used a dtk-computer
to publish a selection of his papers. eral) exact. becomes a better and better with 486i33MHz
mmencing with The Smarandache approximation to S(n) as n becomes processor in Borland's
Function Journal. Vol I. No I, Decem larger and larger.) Turbo Basic. S(n) uoto
ber 1990. ISSN 1053-4792. Problem (ii) For a specified non-null 106 took 2hr SOm'in.
PCWreaders mav have met this func integer m. under what conditions does He completed a great
tion before, in :--iumbers Count -112- Sin) divide t.1.e difference n -m? deal of work on all
July 1992, where a very encouraging Problem (iii) Investigate the possible problems except (vi): details of numer
response was generated. This article integer solutions. (x.y.z) of S(x") + Sty") ical results and conclusions available
[February 1993) is complete in itself so = S(zO) for any u greater than or equal to from Henrv or mvself to interested
don't worry if you have filed the July 1. e.g. examine the solution (5.7,2048) readers. What abo'ut problem (vi)?
issue! It may be thought that those when u = 3.
readers who attempted the previous (It can be proved that an infinity of
problem-set will have an unfair advan solutions exist for anv such n-value.)
tage. However. it must be realised that Compare with Fermat's Theorem reo
no :--iumbers Count problems are com xo+vo = zoo
pletely original so previous work within Problem (iv) Investigate the possibility
a gi ven subject area is al ways a possibil of finding two integers n and k such
ity and the prize is awarded using 'suit that the LOGARITIDvf of S(u') to the
able subjective criteria' anyway, so B.-~SE S(k") is an integer.
please have a go and submit your re Problem (v) Recall that 'Gamma' de
sults. however trivial they mav seem to fined as the limit as n tends to infinity
yourself. - - of ( 1 + 1/2 + 1/3 + 1/4 .... +1/n -log(n)')
Definition For all non-null integers. n. exists, is known as Euler's Constant
the Smarandache Function, Sin), is and is approximately 0.577.
defined to be the smallest integer such Investigate the possible existence of
that (Sin))! (The Factorial Function with 'Samma' defined as the limit as n tends
argument Sin),) is di"isible by n. e.g. to infinity of (1 + 1IS(2) + 1/S(3) .... +1/
S(18) = 6 because 6! is divisible bv 18 Sen) -!og(S(n)).
but l! .... S! are not. • Problem (vi) Find the number ofP;\R
Problem (0) Design and implement an TITIONS of n as the sum of S(m) for
algorithm to generate and store/tabu 2<~. See PCW August 1989 and
late S(n) as a function of n upto a given February 1990 for other problems in
volving PARTITIONS of n.
1
Republi3bed from <Personal Computer World>, No.llB, 403, February
1993 (witb tbe autbor permission), because some of tbe following
research paper3 are referring to tbese open problems_
2
Republished from <Per30nal Computer World>, No.124, 495, August
1993 (witb the author permission) .