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A325178
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Difference between the length of the minimal square containing and the maximal square contained in the Young diagram of the integer partition with Heinz number n.
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14
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0, 0, 1, 1, 2, 1, 3, 2, 0, 2, 4, 2, 5, 3, 1, 3, 6, 1, 7, 2, 2, 4, 8, 3, 1, 5, 1, 3, 9, 1, 10, 4, 3, 6, 2, 2, 11, 7, 4, 3, 12, 2, 13, 4, 1, 8, 14, 4, 2, 1, 5, 5, 15, 2, 3, 3, 6, 9, 16, 2, 17, 10, 2, 5, 4, 3, 18, 6, 7, 2, 19, 3, 20, 11, 1, 7, 3, 4, 21, 4, 2, 12
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OFFSET
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1,5
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COMMENTS
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The maximal square contained in the Young diagram of an integer partition is called its Durfee square, and its length is the rank of the partition.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
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REFERENCES
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Richard P. Stanley, Enumerative Combinatorics, Volume 2, Cambridge University Press, 1999, p. 289.
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LINKS
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FORMULA
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EXAMPLE
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The partition (3,3,2,1) has Heinz number 150 and diagram
o o o
o o o
o o
o
containing maximal square
o o
o o
and contained in minimal square
o o o o
o o o o
o o o o
o o o o
so a(150) = 4 - 2 = 2.
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MATHEMATICA
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durf[n_]:=Length[Select[Range[PrimeOmega[n]], Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]][[#]]>=#&]];
codurf[n_]:=If[n==1, 0, Max[PrimeOmega[n], PrimePi[FactorInteger[n][[-1, 1]]]]];
Table[codurf[n]-durf[n], {n, 100}]
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CROSSREFS
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Cf. A001222, A046660, A051924, A056239, A061395, A093641, A096771, A115994, A243055, A257990, A263297, A325192, A325195.
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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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