Journal of Integer Sequences, Vol. 27 (2024), Article 24.7.1 |

Institute of Mathematics

University of the Philippines Diliman

Diliman, Quezon City 1101

Philippines

**Abstract:**

Define the minimal excludant of an overpartition , denoted
, to be the smallest positive integer that is not a part of the non-overlined parts of . For a positive integer , the function
is the sum of the minimal excludants over all overpartitions of . In this paper, we prove that the
equals the number of partitions of into distinct parts using three colors. We also provide an asymptotic formula for
and show that
is almost always even and is odd exactly when is a triangular number. Moreover, we generalize
using the least -gaps, denoted
, defined as the smallest part of the non-overlined parts of the overpartition appearing less than times. Similarly, for a positive integer , the function
is the sum of the least -gaps over all overpartitions of . We derive a generating function and an asymptotic formula for
. Lastly, we study the arithmetic density of
modulo , where
.

(Concerned with sequences A001936 A022568.)

Received October 20 2023; revised versions received November 6 2023; August 15 2024;
August 16 2024.
Published in *Journal of Integer Sequences*,
August 16 2024.

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