cp's OEIS Frontend

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).

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

First differs from A323304 in lacking 216. First differs from A106543 in having 144.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose omega-sequence has repeated parts. The enumeration of these partitions by sum is given by A325285.

We define the omega-sequence of n (row n of A323023) to have length A323014(n) = adjusted frequency depth of n, and the k-th term is Omega(red^{k-1}(n)), where Omega = A001222 and red^{k} is the k-th functional iteration of red = A181819, defined by red(n = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of n. For example, we have 180 -> 18 -> 6 -> 4 -> 3, so the omega-sequence of 180 is (5,3,2,2,1), which has repeated parts, so 180 is in the sequence.