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A300789
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Heinz numbers of integer partitions whose Young diagram can be tiled by dominos.
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6
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1, 3, 4, 7, 9, 10, 12, 13, 16, 19, 21, 22, 25, 27, 28, 29, 34, 36, 37, 39, 40, 43, 46, 48, 49, 52, 53, 55, 57, 61, 62, 63, 64, 70, 71, 75, 76, 79, 81, 82, 84, 85, 87, 88, 89, 90, 91, 94, 100, 101, 107, 108, 111, 112, 113, 115, 116, 117, 118, 121, 129, 130, 131
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OFFSET
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1,2
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COMMENTS
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The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
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.
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LINKS
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EXAMPLE
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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).
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MAPLE
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a:= proc(n) option remember; local k; for k from 1+
`if`(n=1, 0, a(n-1)) while (l-> add(`if`(l[i]::odd,
(-1)^i, 0), i=1..nops(l))<>0)(sort(map(i->
numtheory[pi](i[1])$i[2], ifactors(k)[2]))) do od; k
end:
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MATHEMATICA
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primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Total[(-1)^Flatten[Position[primeMS[#], _?OddQ]]]===0&] (* Conjectured *)
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CROSSREFS
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Cf. A000712, A000898, A001405, A004003, A045931, A097613, A099390, A299926, A300056, A300060, A300787, A300788, A304662.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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