ARITHMETIC PROPERTIES SATISFIED BY A RECENT INTEGER PARTITION FUNCTION OF DOMBOS

Abstract

The set of integer partitions of n , wherein the parts are either divisible by 4 or congruent to

, arose in a natural way in work of Dombos [‘A new grounded partition identity of type

D 4 ( 3 ) $D_4^{(3)}$ upper D 4 Superscript left parenthesis 3 right parenthesis

’, SIGMA 22 (2026), Article no. 042]. Denote the function that counts the number of such partitions of n by

dp ( n ) $\mathit {dp}(n)$ italic dp left parenthesis n right parenthesis

. Using elementary generating function manipulations and classical q -series results, we prove several congruences satisfied by

dp ( n ) $\mathit {dp}(n)$ italic dp left parenthesis n right parenthesis

. As an example, we prove that, for all

α ≥ 1 $\alpha \geq 1$ alpha greater than or equals 1

and all

n ≥ 0 $n \geq 0$ n greater than or equals 0

,

d p ( 3 2 α + 1 n + 7 ⋅ 9 α + 1 4 ) ≡ 0 ( mod 3 ) . $$ \begin{align*} {\it dp} \bigg( 3^{2\alpha + 1}n + \frac{7 \cdot 9^\alpha + 1}{4} \bigg) \equiv 0 \pmod{3}. \end{align*} $$