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A323585 Third Moebius transform of A000219. Number of plane partitions of n whose multiset of rows is aperiodic and whose multiset of columns is also aperiodic and whose parts are relatively prime.

%I A323585 #7 Jan 22 2019 16:42:11
%S A323585 1,1,0,3,7,21,30,83,129,267,428,856,1332,2482,3909,6798,10853,18331,
%T A323585 28665,47327,73829,118527,183898,290780,446508,695964,1061290,1631829,
%U A323585 2470970,3759609,5646952,8512306,12700005,18972387,28120953,41690725,61392966,90379781
%N A323585 Third Moebius transform of A000219. Number of plane partitions of n whose multiset of rows is aperiodic and whose multiset of columns is also aperiodic and whose parts are relatively prime.
%C A323585 A multiset is aperiodic if its multiplicities are relatively prime.
%H A323585 Alois P. Heinz, <a href="/A323585/b323585.txt">Table of n, a(n) for n = 0..10000</a>
%F A323585 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 second is A323584.
%e A323585 The a(4) = 7 plane partitions with aperiodic multisets of rows and columns and relatively prime parts:
%e A323585   31   211
%e A323585 .
%e A323585   3   21   111
%e A323585   1   1    1
%e A323585 .
%e A323585   2   11
%e A323585   1   1
%e A323585   1   1
%e A323585 The same for a(5) = 21:
%e A323585   41   32   311   221   2111
%e A323585 .
%e A323585   4   3   31   21   22   21   211   111   1111
%e A323585   1   2   1    2    1    11   1     11    1
%e A323585 .
%e A323585   3   2   21   11   111
%e A323585   1   2   1    11   1
%e A323585   1   1   1    1    1
%e A323585 .
%e A323585   2   11
%e A323585   1   1
%e A323585   1   1
%e A323585   1   1
%t A323585 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A323585 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A323585 ptnplane[n_]:=Union[Map[Reverse@*primeMS,Join@@Permutations/@facs[n],{2}]];
%t A323585 Table[Sum[Length[Select[ptnplane[Times@@Prime/@y],And[GCD@@Length/@Split[#]==1,GCD@@Length/@Split[Transpose[PadRight[#]]]==1,And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&]],{y,Select[IntegerPartitions[n],GCD@@#==1&]}],{n,10}]
%Y A323585 Cf. A000219, A000837, A003293, A100953, A300275, A303546, A320802, A321390, A323584, A323587.
%K A323585 nonn
%O A323585 0,4
%A A323585 _Gus Wiseman_, Jan 19 2019