On
an unsolved question about the Smarandache Square-Partial-Digital Subsequence
Felice
Russo
Micron Technology Italy
Avezzano (Aq) – Italy
Abstract
In this note we report the solution
of an unsolved question on Smarandache Square-Partial-Digital Subsequence. We
have found it by extesive computer search.
Some new questions about
palindromic numbers and prime numbers in SSPDS are posed too.
Introduction
The Smarandache Square-Partial-Digital
Subsequence (SSPDS) is the sequence of square integers which admits a partition
for which each segment is a square integer [1],[2],[3].
The first terms of the sequence
follow:
49, 144, 169, 361, 441, 1225,
1369, 1444, 1681, 1936, 3249, 4225, 4900, 11449, 12544, 14641, 15625, 16900 …
or
7, 12, 13, 19, 21, 35, 37, 38, 41,
44, 57, 65, 70, 107, 112, 121, 125, 130, 190, 191, 204, 205, 209, 212, 223, 253
…
reporting the value of n^2 that
can be partitioned into two or more numbers that are also squares (A048653)
[5].
Differently from the sequences
reported in [1], [2] and [3] the proposed ones don’t contain terms that admit 0
as partition. In fact as reported in [4] we don’t consider the number zero a
perfect square.
So, for example, the term 256036
and the term 506 respectively, are not reported in the above sequences because
the partion 256/0/36 contains the number zero.
L. Widmer explored some properties
of SSPDS’s and posed the following question [2]:
Is there a sequence of three or
more consecutive integers whose squares are in SPDS?
This note gives an answer to this
question.
Results
A computer code has been written
in Ubasic Rev. 9.
After about three week of work
only a solution for three consecutive integers has been found. Those consecutive
integers are: 12225, 12226,12227.
n |
n^2 |
Partition |
12225 |
149450625 |
1, 4, 9, 4, 50625 |
12226 |
149475076 |
1, 4, 9, 4, 75076 |
12227 |
149499529 |
1, 4, 9, 4, 9, 9, 529 |
No other three consecutive integers
or more have been found for terms in SSPDS up to about 3.3E+9. Below a graph of
distance dn between the terms of sequence A048653 versus n is given; in
particular dn=a(n+1)-a(n) where n is the n-th term of the sequence.
According to the previous results
we are enough confident to offer the following conjecture:
There are no four consecutive integers whose squares are in SSPDS.
New Questions
Starting with the sequence
(A048646), reported above, the following sequence can be created [5] (A048653):
7, 13, 19, 37, 41, 107, 191, 223,
379, 487, 1093, 1201, 1301, 1907, 3019, 3371, 5081, 9041, 9721, 9907……
that we can call "
Smarandache Prime-Square-Partial-Digital-Subsequence " because all the
squares of these primes can be partitioned into two or more numbers that are
also squares.
By looking this sequence the
following questions can be posed:
If we look now at the terms of the
sequence A048653 we discover that two of them are very interesting:
121 and 212
Both numbers are palindromes and
their squares are in SSPDS and palindromes too. In fact 121^2=14641 can be
partitioned as: 1,4,64,4 and 212^2=44944 can be partitioned in five squares
that are also palindromes: 4, 4, 9, 4, 4. These are the only terms found by our
computer search. So the following question arises:
1.
How
many other SSPDS palindromes do exist ?
References
[1] Sylvester Smith, "A Set
of Conjectures on Smarandache Sequences", Bulletin of Pure and
Applied Sciences, (Bombay, India), Vol. 15 E (No. 1), 1996, pp. 101-107.
[2] L.Widmer, Construction of
Elements of the Smarandache Square-Partial-Digital Sequence, Smarandache
Notions Journal, Vol. 8, No. 1-2-3, 1997, 145-146.
[3] C. Dumitrescu and V. Seleacu,
Some notions and questions in Number Theory, Erhus University Press, Glendale,
Arizona, 1994
[4] E. W. Weisstein, CRC Concise
Encyclopedia of Mathematics, CRC Press 1999, p. 1708
[5} N. Sloane, On-line
Encyclopedia of Integer Sequences, http://www.research.att.com/~njas/sequences