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A323306 Heinz numbers of integer partitions that can be arranged into a matrix with equal row-sums and equal column-sums.

%I A323306 #6 Jan 14 2019 18:25:54
%S A323306 1,2,3,4,5,7,8,9,11,13,16,17,19,23,25,27,29,31,32,36,37,41,43,47,49,
%T A323306 53,59,61,64,67,71,73,79,81,83,89,97,100,101,103,107,109,113,121,125,
%U A323306 127,128,131,137,139,149,151,157,163,167,169,173,179,181,191,193
%N A323306 Heinz numbers of integer partitions that can be arranged into a matrix with equal row-sums and equal column-sums.
%C A323306 First differs from A137944 in lacking 120.
%C A323306 The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%H A323306 <a href="/index/He#Heinz">Index entries for sequences related to Heinz numbers</a>
%e A323306 6480 belongs to the sequence because it is the Heinz number of (3,2,2,2,2,1,1,1,1), which can be arranged in the following ways:
%e A323306   [1 1 3] [1 2 2] [1 2 2] [1 3 1] [2 1 2] [2 1 2] [2 2 1] [2 2 1] [3 1 1]
%e A323306   [2 2 1] [1 2 2] [3 1 1] [2 1 2] [1 3 1] [2 1 2] [1 1 3] [2 2 1] [1 2 2]
%e A323306   [2 2 1] [3 1 1] [1 2 2] [2 1 2] [2 1 2] [1 3 1] [2 2 1] [1 1 3] [1 2 2]
%t A323306 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A323306 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A323306 ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
%t A323306 Select[Range[100],!Select[ptnmats[#],And[SameQ@@Total/@#,SameQ@@Total/@Transpose[#]]&]=={}&]
%Y A323306 Complement of A323304.
%Y A323306 Cf. A006052, A007016, A056239, A112798, A120733, A319056, A319066, A321719.
%Y A323306 Cf. A323300, A323302, A323347, A323348, A323349.
%K A323306 nonn
%O A323306 1,2
%A A323306 _Gus Wiseman_, Jan 13 2019