A265847 Number of different quasi-orders with n labeled elements, modulo n.

0, 0, 2, 3, 2, 1, 2, 2, 0, 3, 2, 1, 2, 6, 1, 15, 2, 1, 2

Offset: 1

Author

Altug Alkan, Dec 21 2015

Keywords

• nonn
• more

Comments

Remainder when number of different quasi-orders with n labeled elements is divided by n.

If n is an odd prime, a(n) = 2 because of the fact that A000798(p^k) == k + 1 mod p for all primes p. For k = 1, A000798(p) == 2 mod p for all primes p.

Currently, A000798 has values for n <= 18. However, thanks to A000798(p) == 2 mod p, we know that a(19) = 2.

How is the distribution of other terms such as 1 and 3 in this sequence?

Examples

			a(4) = A000798(4) mod 4 = 355 mod 4 = 3.
a(5) = A000798(5) mod 5 = 6942 mod 5 = 2.
a(6) = A000798(6) mod 6 = 209527 mod 6 = 1.
		

Links

• Muhammet Yasir Kizmaz, On The Number Of Topologies On A Finite Set, arXiv preprint arXiv:1503.08359 [math.NT], 2015.

Crossrefs

Cf. A000798.

Formula

a(A000040(n)) = 2, for n > 1.