|
%I #8 Jan 17 2019 17:18:27
%S 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,
%T 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,
%U 0,72,0,0,0,0,0,0,0,0,126,252,0,0,0,0,0,0
%N Number of ways to arrange the parts of a multiset whose multiplicities are the prime indices of n into a square matrix.
%C 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}.
%F If A056239(n) is a perfect square, a(n) = A318762(n). Otherwise, a(n) = 0.
%e The a(9) = 6 matrices:
%e [1 1] [1 2] [1 2] [2 1] [2 1] [2 2]
%e [2 2] [1 2] [2 1] [1 2] [2 1] [1 1]
%e The a(38) = 9 matrices:
%e [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]
%e [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]
%e [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]
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,Reverse[primeMS[n]]];
%t Table[If[IntegerQ[Sqrt[Total[primeMS[n]]]],Length[Permutations[nrmptn[n]]],0],{n,100}]
%Y The positions of 0's are numbers whose sum of prime indices is not a perfect square (A323527).
%Y The positions of 1's are primes indexed by squares (A323526).
%Y Cf. A008480, A026478, A056239, A103198, A112798, A120732, A318284, A318762, A323519, A323528, A323531.
%K nonn
%O 1,9
%A _Gus Wiseman_, Jan 17 2019
|
