A. David Christopher
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 26, 2020, Number 2, Pages 116–126
DOI: 10.7546/nntdm.2020.26.2.116-126
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A. David Christopher
Department of Mathematics, The American College
Tamil Nadu, India
Abstract
Let be a positive integer and
be a non-negative integer. We define
and to be the number of ways
can be expressed as a difference of two elements from the sequence
. Nyblom found closed expressions for
and
in terms of some restricted number-of-divisors functions. Here we re-establish these two results of Nyblom in a relatively simple way. Along with the other interpretations for
, an expression for
is presented in terms of restricted form of
and
. Also we consider another function due to Nyblom, denoted
, which counts the number of partitions of
with parts in arithmetic progression having common difference
. Nyblom and Evan found a simple expression for
and put
in terms of a divisor-counting functions when
. Here we re-establish Nyblom’s expression for
, and find equinumerous expressions for
when
. Finally, we present the following generalised version of
: given a set of positive integers say,
, we denote by
, the number of ways
can be written as a difference of two elements from the set
. And we express
in terms of partition enumerations when some restrictions are imposed upon the elements of
. We close with the hint that, boundedness of
together with the divergence of
disproves Erdős arithmetic progression conjecture.
Keywords
- Square numbers
- Triangular numbers
- Representation as difference of integers
2010 Mathematics Subject Classification
- 11A67
- 11B34
References
- Cook, R., & Sharp, D. (1995). Sums of arithmetic progressions, Fibonacci Quart., 33, 218–221.
- Christopher, A. D. (2015). Partitions with Fixed Number of Sizes, J. Integer Seq., 18, Article 15.11.5.
- Mason, T. E. (1912). On the representation of an integer as the sum of consecutive integers, Amer. Math. Monthly, 19 (3), 46–50.
- Munagi, A. O. (2010). Combinatorics of integer partitions in arithmetic progression, Integers, 10, 73–82.
- Nyblom, M. A. (2001). On the Representation of the Integers as a Difference of
non-consecutive Triangular numbers, Fibonacci Quart., 39 (3), 256–263. - Nyblom, M. A. (2002). On the Representation of the Integers as a Difference of Squares, Fibonacci Quart., 40 (3), 243–246.
- Nyblom, M. A., & Evans, C. (2003). On the enumeration of partitions with summands in arithmetic progression, Australas. J. Combin., 28, 149–159.
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Cite this paper
Christopher, A. D. (2020). An alternative proof of Nyblom’s results and a generalisation. Notes on Number Theory and Discrete Mathematics, 26 (2), 116-126, DOI: 10.7546/nntdm.2020.26.2.116-126.