a(n) = 2 iff n is prime.
Number of matrices whose entries are 1,...,n, up to row and column permutations. For example, inequivalent representatives of the a(4) = 8 matrices are:
[1 2 3 4]
.
[1 2] [1 2] [1 3] [1 3] [1 4] [1 4]
[3 4] [4 3] [2 4] [4 2] [2 3] [3 2]
.
[1]
[2]
[3]
[4]
(End)
Conjecture: the sequence a(n) taken modulo a positive integer k >= 3 eventually becomes constant equal to 2. For example, the sequence taken modulo 11 is [1, 2, 2, 8, 2, 1, 2, 10, 6, 3, 2, 2, 2, 2, 2, 2, ...]. -
Peter Bala, Aug 08 2025
a(n) is the number of closed binary operations on n labeled points that can be indexed by a product set so that the operation is given by (a,b)(c,d) = (a,d), i.e. the number of rectangular band semigroup operations on n given points, distinguishing between isomorphic operations. -
David Pasino, Nov 14 2025
