Let  and  be any integers. It is well-known that a positive integer X is called an n-Kaprekar number in the base b if there exist integers   such that  and  For example, 45 is a 2-Kaprekar number in the base 10, since  and  In this article, we give the definition of a variant of Kaprekar numbers. Namely, we call a positive integer x “an n-minus-Kaprekar number in the base b” if there exist positive integers A, B with  such that  and  For example, 1078 is a 3-minus-Kaprekar number in the base 10, since and  Then we see a characterization of minus-Kaprekar numbers and how to construct them. Moreover, as an appendix, we see a brief survey of Iannucci’s result on Kaprekar numbers and its application which includes a new viewpoint of characterizations of Mersenne prime numbers.

Kaprekar numbers, Mersenne prime numbers.