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A325133
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Heinz number of the integer partition obtained by removing the inner lining, or, equivalently, the largest hook, of the integer partition with Heinz number n.
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7
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1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 1, 4, 1, 1, 2, 1, 1, 2, 1, 3, 2, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 1, 5, 3, 2, 1, 1, 4, 3, 1, 2, 1, 1, 2, 1, 1, 4, 1, 3, 2, 1, 1, 2, 3, 1, 2, 1, 1, 6, 1, 5, 2, 1, 1, 8, 1, 1, 2, 3, 1, 2, 1, 1, 4, 5, 1, 2, 1, 3, 1, 1, 5, 4, 3, 1, 2, 1, 1, 6
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OFFSET
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1,9
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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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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 is the partition (4,2) with Heinz number 21, so a(715) = 21.
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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, 1, Times@@Prime/@DeleteCases[Most[primeMS[n]]-1, 0]], {n, 100}]
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PROG
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(PARI)
A052126(n) = if(1==n, n, n/vecmax(factor(n)[, 1]));
A064989(n) = { my(f); f = factor(n); if((n>1 && f[1, 1]==2), f[1, 2] = 0); for (i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f) };
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CROSSREFS
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Positions of ones are A093641 (Heinz numbers of hooks). The number of iterations required to reach 1 starting with n is A257990(n).
Cf. A000720, A001222, A046660, A052126, A056239, A061395, A064989, A112798, A243055, A252464, A325134, A325135.
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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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