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

1,9

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}.

Links

Table of n, a(n) for n=1..84.

Formula

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

Example

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]

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}]

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.

Sequence in context: A097047 A331421 A197581 * A166978 A356547 A365956

Adjacent sequences: A323522 A323523 A323524 * A323526 A323527 A323528

Keyword

nonn

Author

Gus Wiseman, Jan 17 2019

Status

approved