A323303 - OEIS

%I #5 Jan 19 2019 08:47:21

%S 1,1,1,2,1,2,1,2,2,2,1,3,1,2,2,3,1,3,1,3,2,2,1,4,2,2,2,3,1,6,1,2,2,2,

%T 2,10,1,2,2,4,1,6,1,3,3,2,1,5,2,3,2,3,1,4,2,4,2,2,1,12,1,2,3,4,2,6,1,

%U 3,2,6,1,10,1,2,3,3,2,6,1,5,3,2,1,12,2,2

%N Number of ways to arrange the prime indices of n into a matrix with equal column-sums.

%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

%e The a(90) = 16 matrix-arrangements of (3,2,2,1) with equal column-sums:

%e   [1 2] [2 1] [2 3] [3 2]

%e   [3 2] [2 3] [2 1] [1 2]

%e .

%e   [1] [1] [1] [2] [2] [2] [2] [2] [2] [3] [3] [3]

%e   [2] [2] [3] [1] [1] [2] [2] [3] [3] [1] [2] [2]

%e   [2] [3] [2] [2] [3] [1] [3] [1] [2] [2] [1] [2]

%e   [3] [2] [2] [3] [2] [3] [1] [2] [1] [2] [2] [1]

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];

%t Table[Length[Select[ptnmats[n],SameQ@@Total/@Transpose[#]&]],{n,100}]

%Y Cf. A006052, A008480, A056239, A112798, A118914, A120733, A319056, A321719.

%Y Cf. A323295, A323300, A323302, A323305, A323306, A323347, A323349.

%K nonn

%O 1,4

%A _Gus Wiseman_, Jan 17 2019