A323525 Number of ways to arrange the parts of a multiset whose multiplicities are the prime indices of n into a square matrix.

1, 1, 0, 0, 0, 0, 1, 0, 6, 4, 0, 12, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 36, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 84, 0, 0, 72, 0, 0, 0, 0, 0, 0, 0, 0, 126, 252, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Gus Wiseman, Jan 17 2019

Keywords

• nonn

Comments

This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

Examples

			The a(9) = 6 matrices:
  [1 1] [1 2] [1 2] [2 1] [2 1] [2 2]
  [2 2] [1 2] [2 1] [1 2] [2 1] [1 1]
The a(38) = 9 matrices:
  [1 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 2] [1 2 1] [2 1 1]
  [1 1 1] [1 1 1] [1 1 1] [1 1 2] [1 2 1] [2 1 1] [1 1 1] [1 1 1] [1 1 1]
  [1 1 2] [1 2 1] [2 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1]
		

Crossrefs

The positions of 0's are numbers whose sum of prime indices is not a perfect square (A323527).

The positions of 1's are primes indexed by squares (A323526).

Cf. A008480, A026478, A056239, A103198, A112798, A120732, A318284, A318762, A323519, A323528, A323531.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,Reverse[primeMS[n]]];
Table[If[IntegerQ[Sqrt[Total[primeMS[n]]]],Length[Permutations[nrmptn[n]]],0],{n,100}]

Formula

If A056239(n) is a perfect square, a(n) = A318762(n). Otherwise, a(n) = 0.