A323303 Number of ways to arrange the prime indices of n into a matrix with equal column-sums.

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

Offset: 1

Author

Gus Wiseman, Jan 17 2019

Keywords

• nonn

Comments

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.

Examples

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

Crossrefs

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

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

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
Table[Length[Select[ptnmats[n],SameQ@@Total/@Transpose[#]&]],{n,100}]