On some congruences for unrestricted singular overpartitions

A specific case of partition called Andrews’ singular overpartitions, enumerated by [Formula: see text], is the number of overpartitions of [Formula: see text] where no summand is divisible by [Formula: see text] and the first occurrence of any summands which is [Formula: see text] can be marked. A number of authors have studied the properties of this partition. On this paper, we define a new partition which is a modification of singular overpartition, where you may chose any one of the summands which is [Formula: see text] to be marked. We enumerate that particular partition by [Formula: see text] and prove some congruences for [Formula: see text] on modulo [Formula: see text] and [Formula: see text], congruences for [Formula: see text] on modulo [Formula: see text], [Formula: see text] and [Formula: see text], and how likely it is for [Formula: see text] to be even.