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A325135
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Size of the integer partition with Heinz number n after its inner lining, or, equivalently, its largest hook, is removed.
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8
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0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 0, 1, 0, 0, 1, 0, 2, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 0, 3, 2, 1, 0, 0, 2, 2, 0, 1, 0, 0, 1, 0, 0, 2, 0, 2, 1, 0, 0, 1, 2, 0, 1, 0, 0, 3, 0, 3, 1, 0, 0, 3, 0, 0, 1, 2, 0, 1, 0, 0, 2, 3, 0, 1, 0, 2, 0, 0, 3, 2, 2, 0, 1, 0, 0, 3
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OFFSET
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1,25
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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).
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LINKS
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FORMULA
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For n > 1:
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EXAMPLE
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The partition with Heinz number 715 is (6,5,3), with diagram
o o o o o o
o o o o o
o o o
which has inner lining
o o
o o o
o o o
or largest hook
o o o o o o
o
o
both of which have complement
o o o o
o o
which has size 6, so a(715) = 6.
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MATHEMATICA
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Table[If[n==1, 0, Total[Most[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]-1]], {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. A000720, A001222, A052126, A056239, A061395, A093641, A112798, A252464, A257990, A325133, A325134.
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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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