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%I #11 May 22 2018 21:18:41
%S 1,3,4,7,9,10,12,13,16,19,21,22,25,27,28,29,34,36,37,39,40,43,46,48,
%T 49,52,53,55,57,61,62,63,64,70,71,75,76,79,81,82,84,85,87,88,89,90,91,
%U 94,100,101,107,108,111,112,113,115,116,117,118,121,129,130,131
%N Heinz numbers of integer partitions whose Young diagram can be tiled by dominos.
%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C This sequence is conjectured to be the Heinz numbers of integer partitions in which the odd parts appear as many times in even as in odd positions.
%H Alois P. Heinz, <a href="/A300789/b300789.txt">Table of n, a(n) for n = 1..20000</a>
%H Solomon W. Golomb, <a href="https://doi.org/10.1016/S0021-9800(66)80033-9">Tiling with polyominoes</a>, Journal of Combinatorial Theory, 1-2 (1966), 280-296.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Domino_tiling">Domino tiling</a>
%e Sequence of integer partitions whose Young diagram can be tiled by dominos begins: (), (2), (11), (4), (22), (31), (211), (6), (1111), (8), (42), (51), (33), (222), (411).
%p a:= proc(n) option remember; local k; for k from 1+
%p `if`(n=1, 0, a(n-1)) while (l-> add(`if`(l[i]::odd,
%p (-1)^i, 0), i=1..nops(l))<>0)(sort(map(i->
%p numtheory[pi](i[1])$i[2], ifactors(k)[2]))) do od; k
%p end:
%p seq(a(n), n=1..100); # _Alois P. Heinz_, May 22 2018
%t primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Select[Range[100],Total[(-1)^Flatten[Position[primeMS[#],_?OddQ]]]===0&] (* Conjectured *)
%Y Cf. A000712, A000898, A001405, A004003, A045931, A097613, A099390, A299926, A300056, A300060, A300787, A300788, A304662.
%K nonn
%O 1,2
%A _Gus Wiseman_, Mar 12 2018
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