Some Problems Concerning The Smarandache Deconstructive Sequence*
Charles Ashbacher Technologies
Hiawatha, IA 52233
The Smarandache Deconstructive Sequence (SDS(n)) of integers is constructed by sequentially repeating the digits 1-9 in the following way:
1, 23, 456, 7891, 23456, 789123, 4567891, 23456789, 123456789, 1234567891, ...
and first appeared in the collection by Smarandache. In a later collection by Kashihara, the question was asked:
How many primes are there in this sequence?
In this article, we will briefly explore that question and raise a few others concerning this sequence. The values of the first thirty elements of this sequence appear in Table 1.
From the list, it seems clear that the trailing digits repeat the pattern,
1, 3, 6, 1, 6, 3, 1, 9, 9, 1, 3, 6, 1, 6, 3, 1, 9, 9, 1, . . .
and it is simple to prove that this is indeed the case. Given the rules used in the construction of this sequence, the remaining columns also have similar patterns. It is also simple to prove that every third element in the sequence is evenly divisible by 3. More specifically, 3 | SDS(n) if and only if 3 | n.
The list contains the eight primes
23, 4567891, 23456789, 1234567891, 23456789123456789, 23456789123456789123, 4567891234567891234567891, 1234567891234567891234567891.
If we do not consider the first element in the list, the percentage of primes is
-- = 0.276.
Given this, admittedly slim, numeric evidence and the regular nature of the digits, the author is confident enough to offer the following conjecture.
Conjecture 1: The Smarandache Deconstructive Sequence contains an infinite number of primes.
Two out of every nine numbers end in 6. In examining the factorizations of these numbers, we see that 456 is divisible by 23 , 23456 by 25, and all others by 27. This prompts the question:
Question 1: Does every even element of the Smarandache Deconstructive Sequence contain at least three instances of the prime 2 as a factor?
Even more specifically,
Question 2: If we form a sequence from the elements of SDS(n) that end in a 6, do the powers of 2 that divide them form a monotonically increasing sequence?
The following is prompted by examining the divisors of the elements of the sequence.
Question 3: Let k be the largest integer such that 3 | n and j the largest integer such that 3 | SDS(n). Is it true that k is always equal to j?
And we close with the question
Question 4: Are there any other patterns of divisibility in this sequence?
* This paper originally appeared in Journal of Recreational Mathematics, Vol. 29, No. 2. Baywood Publishing Company http://www.baywood.com
1. F. Smarandache, Only Problems, Not Solutions, Xiquan Publishing House, Phoenix, Arizona, 1993.
2. K. Kashihara, Comments and Topics on Smarandache Notions and Problems, Erhus University Press, Vail, Arizona, 1996.