cp's OEIS Frontend

A non-normal magic square is a square matrix with row sums, column sums, and both diagonals all equal to d, for some d|n.

First differs from A137944 in lacking 120.

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Also the number of non-normal semi-magic rectangles summing to n with no zeros.

Matrices must be of size m X k where m, k are divisors of n and mk <= n. This implies that a(p) = 3 for p prime, since the only allowable matrices must be of size 1 X 1, 1 X p or p X 1 with only one way to fill in the entries for each matrix size. Similarly, a(p^2) = 6 with additional allowable matrices of sizes 1 X p^2, p^2 X 1 and p X p, again with only one way to fill in the entries for each size. - Chai Wah Wu, Jan 13 2019

First differs from A046682 at a(11) = 28, A046682(11) = 29.

A vertical section is a partial Young diagram with at most one square in each row. For example, a suitable partition (shown as a coloring by positive integers) of the Young diagram of (322) is:

1 2 3

1 2

2 3

Conjecture: a(n) is the number of half-loop-graphical partitions of n. An integer partition is half-loop-graphical if it comprises the multiset of vertex-degrees of some graph with half-loops, where a half-loop is an edge with one vertex, to be distinguished from a full loop, which has two equal vertices.

An unlabeled version of A279789.

The positions of zeros appear to be A048278.

A non-normal semi-magic square is a nonnegative integer matrix with row sums and column sums all equal to d, for some d|n.

Also the number of nonnegative integer square matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, with row sums and column sums all equal to d.

A non-normal semi-magic square multiset partition of weight n is a multiset partition of weight n whose part sizes and vertex degrees are all equal to d, for some d|n.

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Rectangles must be of size m X k where m, k are divisors of n and mk <= n. This implies that a(p) = 2 for p prime, since the only allowable rectangles must be of size 1 X 1 corresponding to the partition (p), or 1 X p or p X 1 corresponding to the partition (1,1,...,1). Similarly, a(p^2) = 3 since the allowable rectangles must be of sizes 1 X 1 (partition (p^2)), 1 X p or p X 1 (partition (p,p,...,p)), 1 X p^2, p^2 X 1 and p X p (partition (1,1,...,1)). - Chai Wah Wu, Jan 14 2019

An integer partitions is uniform if all parts appear with the same multiplicity.

Terms can be computed by the formula: Sum_{d|n} Sum_{i>=1} P(n/d,i) * Sum_{h|i*d} M(i*d/h, i, h, d) where P(n,k) is the number of partitions of n into k distinct parts and M(h,w,r,s) is the number of nonnegative integer h X w matrices up to row permutations with all row sums equal to r and all column sums equal to s. The cases of M(h,w,w,h) and M(n,n,k,k) are enumerated by the arrays A257462 and A257463. - Andrew Howroyd, Feb 04 2022