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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.

%I A322792 #18 Mar 05 2025 22:04:51
%S A322792 2,6,4,30,8,210,36,16,2310,32,30030,900,216,64,510510,128,9699690,
%T A322792 44100,1296,256,223092870,27000,512,6469693230,5336100,7776,1024,
%U A322792 200560490130,2048,7420738134810,901800900,9261000,810000,46656,4096,304250263527210,8192
%N 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.
%C A322792 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).
%e A322792 Triangle begins:
%e A322792            2
%e A322792            6          4
%e A322792           30          8
%e A322792          210         36         16
%e A322792         2310         32
%e A322792        30030        900        216         64
%e A322792       510510        128
%e A322792      9699690      44100       1296        256
%e A322792    223092870      27000        512
%e A322792   6469693230    5336100       7776       1024
%e A322792 Corresponding triangle of integer partitions whose Heinz numbers belong to the triangle begins:
%e A322792   (1)
%e A322792   (21)        (11)
%e A322792   (321)       (111)
%e A322792   (4321)      (2211)      (1111)
%e A322792   (54321)     (11111)
%e A322792   (654321)    (332211)    (222111)    (111111)
%e A322792   (7654321)   (1111111)
%e A322792   (87654321)  (44332211)  (22221111)  (11111111)
%e A322792   (987654321) (333222111) (111111111)
%t A322792 Table[Product[Prime[i]^d,{i,n/d}],{n,12},{d,Divisors[n]}]
%Y A322792 First column is A002110.
%Y A322792 Cf. A000961, A001597, A002110, A007947, A027750, A025487, A047966, A055932, A056239, A072774, A100778, A304250, A322793.
%K A322792 nonn,tabf
%O A322792 1,1
%A A322792 _Gus Wiseman_, Dec 26 2018
%E A322792 Name edited by _Peter Munn_, Mar 05 2025