A323519 - OEIS

%I #11 Feb 23 2019 10:10:45

%S 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,

%T 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,

%U 0,0,1,0,1,0,0,0,0,0,1,0,1,0,1,12,0,0

%N 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.

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

%e The a(60) = 12 matrices:

%e   [2 2] [2 2] [2 3] [2 3] [2 5] [2 5] [3 2] [3 2] [3 5] [5 2] [5 2] [5 3]

%e   [3 5] [5 3] [2 5] [5 2] [2 3] [3 2] [2 5] [5 2] [2 2] [2 3] [3 2] [2 2]

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

%t Table[If[IntegerQ[Sqrt[PrimeOmega[n]]],Length[Permutations[primeMS[n]]],0],{n,100}]

%Y Positions of 0's are A323521.

%Y Positions of 1's are A323520.

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

%K nonn

%O 1,24

%A _Gus Wiseman_, Jan 17 2019