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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. 4
1, 1, 0, 3, 7, 21, 30, 83, 129, 267, 428, 856, 1332, 2482, 3909, 6798, 10853, 18331, 28665, 47327, 73829, 118527, 183898, 290780, 446508, 695964, 1061290, 1631829, 2470970, 3759609, 5646952, 8512306, 12700005, 18972387, 28120953, 41690725, 61392966, 90379781 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

A multiset is aperiodic if its multiplicities are relatively prime.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..10000

FORMULA

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.

EXAMPLE

The a(4) = 7 plane partitions with aperiodic multisets of rows and columns and relatively prime parts:

  31   211

.

  3   21   111

  1   1    1

.

  2   11

  1   1

  1   1

The same for a(5) = 21:

  41   32   311   221   2111

.

  4   3   31   21   22   21   211   111   1111

  1   2   1    2    1    11   1     11    1

.

  3   2   21   11   111

  1   2   1    11   1

  1   1   1    1    1

.

  2   11

  1   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]]}]];

ptnplane[n_]:=Union[Map[Reverse@*primeMS, Join@@Permutations/@facs[n], {2}]];

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}]

CROSSREFS

Cf. A000219, A000837, A003293, A100953, A300275, A303546, A320802, A321390, A323584, A323587.

Sequence in context: A018479 A090504 A018548 * A003585 A108102 A065523

Adjacent sequences:  A323582 A323583 A323584 * A323586 A323587 A323588

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 19 2019

STATUS

approved

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Last modified December 9 09:20 EST 2019. Contains 329877 sequences. (Running on oeis4.)