cp's OEIS Frontend

After 2, can this ever be prime? This is to A001923 Sum k^k, k=1..n, as k^k^k is to k^k.

Here are several different ways of expressing the condition that g*b = b:

b(u, v) = b(gu, gv) for all u, v in S.

The level sets of b are closed under g x g.

The level sets of b are unions of cycles of g x g.

The cycles of g x g are subsets of level sets of b.

b is constant on cycles of g x g.

There is no requirement for g to be an automorphism of b. Given g, the fixed b are determined by simply choosing a value in S for each cycle of g x g. The product b(u, v) is defined to be that constant value for every (u, v) in the cycle.

So the number of degrees of freedom for b is the number of cycles of g x g. How many cycles does g have on S x S? If u is in a c-cycle C and v is in a d-cycle D, then (u, v) is in an lcm(c, d)-cycle and C x D is partitioned into these cycles, so there must be cd/lcm(c, d) of them, which is gcd(c, d).

So letting s_k be the number of k-cycles of g on S for each k from 1 to n, the total number of cycles of g on S x S is the sum on k and j of gcd(k, j) s_k s_j. That's the number of degrees of freedom for b and each degree has valence n, so raise n to that power. Then multiply by the well-known number of permutations of type s, which is n! divided by the factorials of the s_k and by the powers k^s_k. Add this up over all the partitions of n and divide by n!.

Additional comments from Christian G. Bower: This is the number of nonisomorphic n-state relations on a set of n elements. If at the step of raising n to the power, we raised instead some constant m to that power, the formula would give the number of isomorphism classes of m-state relations on an n-element set.

Isomorphisms classes of a set A with two functions f1,f2: A X A -> A.

A001426(n) is the number of commutative semigroups of order n. A001426(n) + A079193(n) + A079196(n) + A079199(n) = A001329(n). 2, 5, 17, 2543 and 241639 are primes.

The number of isomorphism classes of closed binary operations on sets of order <= n. See formulas by Christian G. Bower in A001329 Number of nonisomorphic groupoids with n elements.

In a sense, this measures the increase in combinatorial structures available by dropping the requirement of inverses, and an identity element, in moving from the group axioms to the semigroup axioms. A semigroup is mathematical object defined for a set and a binary operator in which the multiplication operation is associative. No other restrictions are placed on a semigroup; thus a semigroup need not have an identity element and its elements need not have inverses within the semigroup. Other sequences may be derived by considering commutative semigroups and commutative groups, self-converse semigroup, counting idempotents, and the like.

A magma S is called an Abel-Grassmann or AG-groupoid (historically they were also called left almost semigroups, right modular groupoids and left invertive groupoids) if for all a,b,c in S (ab)c = (cb)a.