A322792 Irregular triangle read by rows where T(n,k) = A002110(n/d)^d, where d = A027750(n,k) and A002110(m) is the product of the first m primes.

2, 6, 4, 30, 8, 210, 36, 16, 2310, 32, 30030, 900, 216, 64, 510510, 128, 9699690, 44100, 1296, 256, 223092870, 27000, 512, 6469693230, 5336100, 7776, 1024, 200560490130, 2048, 7420738134810, 901800900, 9261000, 810000, 46656, 4096, 304250263527210, 8192

Offset: 1

Author

Gus Wiseman, Dec 26 2018

Keywords

• nonn
• tabf

Comments

A reordering of A100778 (powers of primorials), these are the Heinz numbers of uniform integer partitions of length n whose union is an initial interval of positive integers. An integer partition is uniform if all parts appear with the same multiplicity. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Examples

			Triangle begins:
           2
           6          4
          30          8
         210         36         16
        2310         32
       30030        900        216         64
      510510        128
     9699690      44100       1296        256
   223092870      27000        512
  6469693230    5336100       7776       1024
Corresponding triangle of integer partitions whose Heinz numbers belong to the triangle begins:
  (1)
  (21)        (11)
  (321)       (111)
  (4321)      (2211)      (1111)
  (54321)     (11111)
  (654321)    (332211)    (222111)    (111111)
  (7654321)   (1111111)
  (87654321)  (44332211)  (22221111)  (11111111)
  (987654321) (333222111) (111111111)
		

Crossrefs

First column is A002110.

Cf. A000961, A001597, A002110, A007947, A027750, A025487, A047966, A055932, A056239, A072774, A100778, A304250, A322793.

Programs

Mathematica

Table[Product[Prime[i]^d,{i,n/d}],{n,12},{d,Divisors[n]}]

Extensions

Name edited by Peter Munn, Mar 05 2025