login
A323307
Number of ways to fill a matrix with the parts of a multiset whose multiplicities are the prime indices of n.
11
1, 1, 2, 4, 2, 6, 3, 12, 18, 12, 2, 36, 4, 10, 20, 72, 2, 60, 4, 40, 60, 24, 3, 120, 80, 14, 360, 120, 4, 240, 2, 240, 42, 32, 70, 720, 6, 27, 112, 480, 2, 210, 4, 84, 420, 40, 4, 1440, 280, 280, 108, 224, 5, 1260, 224, 420, 180, 22, 2, 840, 6, 72, 1680, 2880
OFFSET
1,3
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}.
FORMULA
a(n) = A318762(n) * A000005(A056239(n)).
EXAMPLE
The a(22) = 24 matrices:
[111112] [111121] [111211] [112111] [121111] [211111]
.
[111] [111] [111] [112] [121] [211]
[112] [121] [211] [111] [111] [111]
.
[11] [11] [11] [11] [12] [21]
[11] [11] [12] [21] [11] [11]
[12] [21] [11] [11] [11] [11]
.
[1] [1] [1] [1] [1] [2]
[1] [1] [1] [1] [2] [1]
[1] [1] [1] [2] [1] [1]
[1] [1] [2] [1] [1] [1]
[1] [2] [1] [1] [1] [1]
[2] [1] [1] [1] [1] [1]
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS, facs[n], {2}]), SameQ@@Length/@#&];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]]];
Array[Length[ptnmats[Times@@Prime/@nrmptn[#]]]&, 30]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 13 2019
STATUS
approved