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A323584 Second Moebius transform of A000219. Number of plane partitions of n whose multiset of rows is aperiodic and whose multiset of columns is also aperiodic.

%I A323584 #9 Jan 22 2019 16:45:09
%S A323584 1,1,1,4,8,22,34,84,137,271,450,857,1373,2483,3993,6823,10990,18332,
%T A323584 28966,47328,74286,118614,184755,290781,448010,695986,1063773,1632100,
%U A323584 2474970,3759610,5654233,8512307,12710995,18973247,28139285,41690830,61423271,90379782
%N A323584 Second Moebius transform of A000219. Number of plane partitions of n whose multiset of rows is aperiodic and whose multiset of columns is also aperiodic.
%C A323584 A multiset is aperiodic if its multiplicities are relatively prime.
%C A323584 Also the number of plane partitions of n whose multiset of rows is aperiodic and whose parts are relatively prime.
%H A323584 Alois P. Heinz, <a href="/A323584/b323584.txt">Table of n, a(n) for n = 0..10000</a>
%F A323584 The Moebius transform T of a sequence q is T(q)(n) = Sum_{d|n} mu(n/d) * q(d) where mu = A008683. The first Moebius transform of A000219 is A300275 and the third is A323585.
%e A323584 The a(4) = 8 plane partitions with aperiodic multisets of rows and columns:
%e A323584   4   31   211
%e A323584 .
%e A323584   3   21   111
%e A323584   1   1    1
%e A323584 .
%e A323584   2   11
%e A323584   1   1
%e A323584   1   1
%e A323584 The a(4) = 8 plane partitions with aperiodic multiset of rows and relatively prime parts:
%e A323584   31   211   1111
%e A323584 .
%e A323584   3   21   111
%e A323584   1   1    1
%e A323584 .
%e A323584   2   11
%e A323584   1   1
%e A323584   1   1
%t A323584 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A323584 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A323584 ptnplane[n_]:=Union[Map[Reverse@*primeMS,Join@@Permutations/@facs[n],{2}]];
%t A323584 Table[Sum[Length[Select[ptnplane[Times@@Prime/@y],And[GCD@@Length/@Split[#]==1,And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&]],{y,Select[IntegerPartitions[n],GCD@@#==1&]}],{n,10}]
%Y A323584 Cf. A000219, A000837, A003293, A100953, A300275, A303546, A320802, A321390, A323585, A323587.
%K A323584 nonn
%O A323584 0,4
%A A323584 _Gus Wiseman_, Jan 19 2019