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A325166
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Size of the internal portion of the integer partition with Heinz number n.
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14
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0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 2, 0, 0, 2, 0, 1, 2, 1, 0, 1, 2, 1, 2, 1, 0, 3, 0, 0, 2, 1, 3, 2, 0, 1, 2, 1, 0, 3, 0, 1, 3, 1, 0, 1, 3, 3, 2, 1, 0, 3, 3, 1, 2, 1, 0, 3, 0, 1, 3, 0, 3, 3, 0, 1, 2, 4, 0, 2, 0, 1, 4, 1, 4, 3, 0, 1, 3, 1, 0, 3, 3, 1, 2, 1, 0, 4, 4, 1, 2, 1, 3, 1, 0, 4, 3, 3, 0, 3, 0, 1, 5
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OFFSET
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1,15
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COMMENTS
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The internal portion of an integer partition consists of all squares in the Young diagram that have a square both directly below and directly to the right.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
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LINKS
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FORMULA
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EXAMPLE
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The partition with Heinz number 7865 is (6,5,5,3), with diagram
o o o o o o
o o o o o
o o o o o
o o o
with internal portion
o o o o o
o o o o
o o o
of size 12, so a(7865) = 12.
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[If[n==1, 0, Total[primeMS[n]]-Max[primeMS[n]]-Length[primeMS[n]]+Length[Union[primeMS[n]]]], {n, 100}]
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PROG
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(PARI)
A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
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CROSSREFS
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Cf. A001221, A001222, A052126, A056239, A061395, A064989, A065770, A112798, A252464, A257990, A297113, A325133, A325135, A325167, A325169.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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