Some results about four Smarandache U-product 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 U-product sequence.

Let , be a positive integer sequence. Then a U-sequence is defined as follows:

In this paper differently from [1], we will call this sequence a U-sequence of the first kind because we will introduce for the first time a U-sequence 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:

1. How many terms are prime?
2. Is the Smarandache Series convergent?
3. Is the Smarandache Infinite product convergent?
4. Is the Smarandache Simple Continued Fractions convergent?
5. Is the Smarandache Continued Radicals convergent?

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 1st kind 12/456=0.026 0.7288315379….. 0 2.1989247812…. 2.3666079803…. Square 2nd kind 1/463=0.0021 0.3301888340…. 1.8143775546…. Cubic 1st kind @ 0.6157923201….. 0 2.1110542477…. 2.6904314681…. Cubic 2nd kind @ 0.1427622842…. 2.2446613806…. Factorial 1st kind 5/70=0.071 0.9137455924….. 0 2.3250021620…. 2.2332152218…. Factorial 2nd kind 2/66=0.033 0.9166908563…. 1.6117607295…. Palprime 1st kind 10/363=0.027 0.5136249121….. 0 3.1422019345…. 2.5932060878…. Palprime 2nd 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 (2nd kind), cubic product (2nd kind) and factorial product (2nd 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 digit’s number.

 n d Square 1st 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 2nd kind 2 1 Cubic 1st kind 1 1 Cubic 2nd kind 2 1 Factorial 1st kind 1/2/3/7/14 1/1/2/125/65 Factroial 2nd kind 3/7 2/12.. Palprime 1st kind 1/2/3/4/5/7/10/19/57/234 1/1/2/3/4/8/15/39/198/1208 Palprime 2nd 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 (1st and 2nd kind), Smarandache Factorial product sequence (1st and 2nd kind) and Smarandache Palprime product sequence (1st and 2nd 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, 237-240.

[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. 1-2, 1998, 40-42

[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. 1-2, 1998, 97-97