A323519 a(n) is the number of ways to fill a square matrix with the multiset of prime factors of n, if the number of prime factors (counted with multiplicity) is a perfect square, and a(n) = 0 otherwise.

1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 4, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 6, 1, 0, 0, 4, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 4, 0, 4, 0, 0, 1, 12, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 12, 0, 0

Offset

1,24

Links

Table of n, a(n) for n=1..86.

Formula

If A001222(n) is a perfect square, then a(n) = A008480(n). Otherwise, a(n) = 0.

Example

The a(60) = 12 matrices:
  [2 2] [2 2] [2 3] [2 3] [2 5] [2 5] [3 2] [3 2] [3 5] [5 2] [5 2] [5 3]
  [3 5] [5 3] [2 5] [5 2] [2 3] [3 2] [2 5] [5 2] [2 2] [2 3] [3 2] [2 2]

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[If[IntegerQ[Sqrt[PrimeOmega[n]]], Length[Permutations[primeMS[n]]], 0], {n, 100}]

Crossrefs

Positions of 0's are A323521.

Positions of 1's are A323520.

Cf. A000290, A008480, A026478, A056239, A089299, A103198, A112798, A120732, A323433, A323525, A323531.

Sequence in context: A348250 A289110 A287287 * A333979 A276580 A331437

Adjacent sequences: A323516 A323517 A323518 * A323520 A323521 A323522

Keyword

nonn

Author

Gus Wiseman, Jan 17 2019

Status

approved