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A367581
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Sum of the multiset multiplicity kernel (in which each multiplicity becomes the least element of that multiplicity) of the prime indices of n.
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14
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0, 1, 2, 1, 3, 2, 4, 1, 2, 2, 5, 3, 6, 2, 4, 1, 7, 3, 8, 4, 4, 2, 9, 3, 3, 2, 2, 5, 10, 3, 11, 1, 4, 2, 6, 2, 12, 2, 4, 4, 13, 3, 14, 6, 5, 2, 15, 3, 4, 4, 4, 7, 16, 3, 6, 5, 4, 2, 17, 5, 18, 2, 6, 1, 6, 3, 19, 8, 4, 3, 20, 3, 21, 2, 5, 9, 8, 3, 22, 4, 2, 2
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OFFSET
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1,3
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COMMENTS
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A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
We define the multiset multiplicity kernel MMK(m) of a multiset m by the following property, holding for all distinct multiplicities k >= 1. If S is the set of elements of multiplicity k in m, then min(S) has multiplicity |S| in MMK(m). For example, MMK({1,1,2,2,3,4,5}) = {1,1,3,3,3}, and MMK({1,2,3,4,5,5,5,5}) = {1,1,1,1,5}. As an operation on multisets, MMK is represented by A367579, and as an operation on their Heinz numbers, it is represented by A367580.
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LINKS
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FORMULA
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a(n^k) = a(n) for all positive integers n and k.
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EXAMPLE
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The multiset multiplicity kernel of {1,2,2,3} is {1,1,2}, so a(90) = 4.
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MATHEMATICA
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mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q, Count[q, #]==i&], {i, mts}]]];
Table[Total[mmk[PrimePi/@Join@@ConstantArray@@@FactorInteger[n]]], {n, 100}]
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CROSSREFS
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Positions of 1's are A000079 without 1.
Positions of first appearances are A008578.
The triangle for this rank statistic is A367582.
Cf. A000720, A005117, A051904, A055396, A061395, A071625, A072774, A130091, A175781, A367584, A367585.
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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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