Some
results about four Smarandache Uproduct sequences
Felice
Russo
Micron Technology Italy
Avezzano (Aq) Italy
Abstract
In this paper four Smarandache product sequences have
been studied: Smarandache Square product sequence, Smarandache Cubic product
sequence, Smarandache Factorial product sequence and Smarandache Palprime
product sequence. In particular the number of primes, the convergence value for
Smarandache Series, Smarandache Continued Fractions, Smarandache Infinite
product of the mentioned sequences has been calculated utilizing the Ubasic
software package. Moreover for the first time the notion of Smarandache
Continued Radicals has been introduced. One conjecture about the number of
primes contained in these sequences and new questions are posed too.
Introduction
In [1] Iacobescu describes the so called
Smarandache Uproduct sequence.
Let , be a
positive integer sequence. Then a Usequence is defined as follows:
In this paper differently from [1], we will call
this sequence a Usequence of the first kind because we will introduce for the
first time a Usequence of the second kind defined as follows:
In this paper we will discuss about the
"Square product", "Cubic product", "Factorial
product" and "Primorial product" sequences. In particular we
will analyze the question posed by Iacobescu in [1] on the number of primes
contained in those sequences. We will also analyze the convergence values of
the Smarandache Series [2], Infinite product [3], Simple Continued Fractions
[4] of the four sequences. Moreover for the first time we will introduce the
notion of Smarandache Continued Radicals and we will analyse the convergence of
sequences reported above.
Sequences details
o Smarandache square product sequence of the
first and second kind.
In this case the sequence is given by:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144
that is the square of n. The first 20 terms of the
sequence () both the first and second
kind are reported in the table below:
Smarandache Square product sequence (first kind) 
Smarandache Square product sequence (second kind) 
2 
0 
5 
3 
37 
35 
577 
575 
14401 
14399 
518401 
518399 
25401601 
25401599 
1625702401 
1625702399 
131681894401 
131681894399 
13168189440001 
13168189439999 
1593350922240001 
1593350922239999 
229442532802560001 
229442532802559999 
38775788043632640001 
38775788043632639999 
7600054456551997440001 
7600054456551997439999 
1710012252724199424000001 
1710012252724199423999999 
437763136697395052544000001 
437763136697395052543999999 
126513546505547170185216000001 
126513546505547170185215999999 
40990389067797283140009984000001 
40990389067797283140009983999999 
14797530453474819213543604224000001 
14797530453474819213543604223999999 
5919012181389927685417441689600000001 
5919012181389927685417441689599999999 
o Smarandache cubic product sequence of the
first and second kind.
In this case the sequence is given by:
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331,
1728
that is the cube of n. Here the first 17 terms for
the sequence of the
first and second kind.
Smarandache Cubic product sequence (first kind) 
Smarandache Cubic product sequence (second kind) 
2 
0 
9 
7 
217 
215 
13825 
13823 
1728001 
1727999 
373248001 
373247999 
128024064001 
128024063999 
65548320768001 
65548320767999 
47784725839872001 
47784725839871999 
47784725839872000001 
47784725839871999999 
63601470092869632000001 
63601470092869631999999 
109903340320478724096000001 
109903340320478724095999999 
241457638684091756838912000001 
241457638684091756838911999999 
662559760549147780765974528000001 
662559760549147780765974527999999 
2236139191853373760085164032000000001 
2236139191853373760085164031999999999 
9159226129831418921308831875072000000001 
9159226129831418921308831875071999999999 
44999277975861761160390291002228736000000001 
44999277975861761160390291002228735999999999 
o Smarandache factorial product sequence of
the first and second kind.
In this case the sequence is given by:
1, 2, 6, 24, 120, 720, 5040, 40320, 362880
.
that is the factorial of n. The first 13 terms of
the sequence of the
first and second kind follow.
Smarandache Factorial product sequence (first kind) 
Smarandache Factorial product sequence (second kind) 
2 
0 
3 
1 
13 
11 
289 
287 
34561 
34559 
24883201 
24883199 
125411328001 
125411327999 
5056584744960001 
5056584744959999 
1834933472251084800001 
1834933472251084799999 
6658606584104736522240000001 
6658606584104736522239999999 
265790267296391946810949632000000001 
265790267296391946810949631999999999 
127313963299399416749559771247411200000000001 
127313963299399416749559771247411199999999999 
792786697595796795607377086400871488552960000000000001 
792786697595796795607377086400871488552959999999999999 
o Smarandache primorial product sequence of
the first and second kind.
In this case the sequence is given by:
2, 3, 5, 7, 11, 101, 121,131, 151, 181, 191, 313,
353, 353, 373
that is the sequence of palindromic primes. Below
the first 17 terms of the sequence of the first and second kind.
Smarandache Palprime product sequence (first kind) 
Smarandache Palprime product sequence (second kind) 
3 
1 
7 
5 
31 
29 
211 
209 
2311 
2309 
233311 
233309 
28230511 
28230509 
3698196811 
3698196809 
558427718311 
558427718309 
101075417014111 
101075417014109 
19305404649695011 
19305404649695009 
6042591655354538131 
6042591655354538129 
2133034854340151959891 
2133034854340151959889 
795622000668876681038971 
795622000668876681038969 
304723226256179768837925511 
304723226256179768837925509 
221533785488242691945171845771 
221533785488242691945171845769 
167701075614599717802495087247891 
167701075614599717802495087247889 
Results
For all above sequences the following qestions have
been studied:
For this purpose the software package Ubasic Rev. 9
has been utilized. In particular for the item n. 1, a strong pseudoprime test
code has been written [5]. Moreover, as already mentioned above, the item 5 has
been introduced for the first time; a Smarandache Continued Radicals is defined
as follows:
where a(n) is the nth term of a Smarandache
sequence. Here below a summary table of the obtained results:

# Primes 
SS_cv 
SIP_cv 
SSCF_cv 
SCR_cv 
Square 1^{st} kind 
12/456=0.026 
0.7288315379 .. 
0 
2.1989247812 . 
2.3666079803 . 
Square 2^{nd} kind 
1/463=0.0021 
0.3301888340 . 
1.8143775546 . 

Cubic 1^{st} kind 
@ 
0.6157923201 .. 
0 
2.1110542477 . 
2.6904314681 . 
Cubic 2^{nd} kind 
@ 
0.1427622842 . 
2.2446613806 . 

Factorial 1^{st} kind 
5/70=0.071 
0.9137455924 .. 
0 
2.3250021620 . 
2.2332152218 . 
Factorial 2^{nd} kind 
2/66=0.033 
0.9166908563 . 
1.6117607295 . 

Palprime 1^{st} kind 
10/363=0.027 
0.5136249121 .. 
0 
3.1422019345 . 
2.5932060878 . 
Palprime 2^{nd} kind 
9/363=0.024 
1.2397048573 .. 
0 
1.1986303614 . 
2.1032632883 . 
Legend:
# primes (Number of primes/number of sequence terms
checked)
SS_cv (Smarandache Series convergence value)
SIP_cv (Smarandache Infinite Product convergence
value)
SSCF_cv (Smarandache Simple Continued Fractions
convergence value)
SCR _cv (Smarandache Continued Radicals convergence
value)
@ (This sequence contain only one prime as proved
by M. Le and K. Wu [6] )
About the items 2,3,4 and 5 according to these results
the answer is: yes, all the analyzed sequences converge except the Smarandache
Series and the Smarandache Infinite product for the square product (2^{nd}
kind), cubic product (2^{nd} kind) and factorial product (2^{nd}
kind). In particular notice the nice result obtained with the convergence of
Smarandache Simple Continued Fractions of Smarandache palprime product sequence
of the first kind.
The value of convergence is roughly with the first two decimal digits
correct.
Analogously for the cubic product sequence of the
second kind the simple continued fraction converge roughly to , while for the factorial product
sequence of the second kind the continued radical converge roughly (two first decimal
digits correct) to the golden ratio , that is:
About the item 1, the following table reports the
values of n in the sequence that generate a strong pseudoprime number and its
digits number.

n 
d 
Square 1^{st} kind 
1/2/3/4/5/9/10/11/1324/65/76 
1/1/2/3/5/12/14/16/20/48/182/223 
Square 2^{nd} kind 
2 
1 
Cubic 1^{st} kind 
1 
1 
Cubic 2^{nd} kind 
2 
1 
Factorial 1^{st} kind 
1/2/3/7/14 
1/1/2/125/65 
Factroial 2^{nd} kind 
3/7 
2/12.. 
Palprime 1^{st} kind 
1/2/3/4/5/7/10/19/57/234 
1/1/2/3/4/8/15/39/198/1208 
Palprime 2^{nd} kind 
2/3/4/5/7/10/19/57/234 
1/2/3/4/8/15/39/198/1208 
Please note that the primes in the sequence of
palprime of the first and second kind generate pairs of twin primes. The first
ones follow:
(3,5) (5,7) (29,31) (209,211)
(2309,2311) (28230509,28230511) (101075417014109,101075417014111)
..
Due to the fact that the percentage of primes found
is very small and that according to Prime Number Theorem, the probability that
a randomly chosen number of size n is prime decreases as 1/d (where d is the
number of digits of n) we are enough confident to pose the following
conjecture:
The
number of primes contained in the Smarandache Square product sequence (1^{st}
and 2^{nd} kind), Smarandache Factorial product sequence (1^{st}
and 2^{nd} kind) and Smarandache Palprime product sequence (1^{st}
and 2^{nd} kind) is finite.
New Questions
Is there any
Smarandache sequence whose SS, SIP, SSCF and SCR converge to some known
mathematical constants?
Are all the
estimated convergence values irrational or trascendental?
Is there for
each prime inside the Smarandache Palprime product sequence of the second kind
the correspondent twin prime in the Smarandache Palprime product sequence of
the first kind?
Are there any
two Smarandache sequences a(n) and b(n) whose Smarandache Infinite Product
ratio converge to some value k different from zero?
Is there any
Smarandache sequence a(n) such that:
For the four
sequences of first kind a(n), study:
where R(a(n)) is the reverse of a(n). (For example
if a(n)=17 then R(a(n))=71 and so on).
References
[1] F. Iacobescu, Smarandache partition type and
other sequences, Bull. Pure Appl. Sci. Sec. E16(1997), No. 2, 237240.
[2] C. Ashbacher, Smarandache Series convergence, to
appear
[3] See
http://www.gallup.unm.edu/~smarandache/product .txt
[4] C. Zhong, On Smarandache Continued fractions,
Smarandache Notions Journal, Vol. 9, No. 12, 1998, 4042
[5] D.M. Bressoud, Factorization and primality testing,
Springer Verlag, 1989, p. 77
[6] M. Le and K. Wu , The primes in the Smarandache
Power product Sequence, Smarandache Notions Journal, Vol. 9, No. 12, 1998,
9797