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A367579 Irregular triangle read by rows where row n is the multiset multiplicity kernel (MMK) of the multiset of prime indices of n.

%I A367579 #16 Nov 27 2023 07:57:30
%S A367579 1,2,1,3,1,1,4,1,2,1,1,5,1,2,6,1,1,2,2,1,7,1,2,8,1,3,2,2,1,1,9,1,2,3,
%T A367579 1,1,2,1,4,10,1,1,1,11,1,2,2,1,1,3,3,1,1,12,1,1,2,2,1,3,13,1,1,1,14,1,
%U A367579 5,2,3,1,1,15,1,2,4,1,3,2,2,1,6,16,1,2
%N A367579 Irregular triangle read by rows where row n is the multiset multiplicity kernel (MMK) of the multiset of prime indices of n.
%C A367579 Row n = 1 is empty.
%C A367579 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.
%C A367579 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}.
%C A367579 Note: I chose the word 'kernel' because, as with A007947 and A304038, MMK(m) is constructed using the same underlying elements as m and has length equal to the number of distinct elements of m. However, it is not necessarily a submultiset of m.
%F A367579 For all positive integers n and k, row n^k is the same as row n.
%e A367579 The first 45 rows:
%e A367579      1: {}      16: {1}       31: {11}
%e A367579      2: {1}     17: {7}       32: {1}
%e A367579      3: {2}     18: {1,2}     33: {2,2}
%e A367579      4: {1}     19: {8}       34: {1,1}
%e A367579      5: {3}     20: {1,3}     35: {3,3}
%e A367579      6: {1,1}   21: {2,2}     36: {1,1}
%e A367579      7: {4}     22: {1,1}     37: {12}
%e A367579      8: {1}     23: {9}       38: {1,1}
%e A367579      9: {2}     24: {1,2}     39: {2,2}
%e A367579     10: {1,1}   25: {3}       40: {1,3}
%e A367579     11: {5}     26: {1,1}     41: {13}
%e A367579     12: {1,2}   27: {2}       42: {1,1,1}
%e A367579     13: {6}     28: {1,4}     43: {14}
%e A367579     14: {1,1}   29: {10}      44: {1,5}
%e A367579     15: {2,2}   30: {1,1,1}   45: {2,3}
%t A367579 mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];
%t A367579 Table[mmk[PrimePi/@Join@@ConstantArray@@@If[n==1, {},FactorInteger[n]]], {n,100}]
%Y A367579 Indices of empty and singleton rows are A000961.
%Y A367579 Row lengths are A001221.
%Y A367579 Depends only on rootless base A052410, see A007916.
%Y A367579 Row minima are A055396.
%Y A367579 Rows have A071625 distinct elements.
%Y A367579 Indices of constant rows are A072774.
%Y A367579 Indices of strict rows are A130091.
%Y A367579 Rows have Heinz numbers A367580.
%Y A367579 Row sums are A367581.
%Y A367579 Row maxima are A367583, opposite A367587.
%Y A367579 Index of first row with Heinz number n is A367584.
%Y A367579 Sorted row indices of first appearances are A367585.
%Y A367579 Indices of rows of the form {1,1,...} are A367586.
%Y A367579 Agrees with sorted prime signature at A367683, counted by A367682.
%Y A367579 A submultiset of prime indices at A367685, counted by A367684.
%Y A367579 A007947 gives squarefree kernel.
%Y A367579 A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
%Y A367579 A124010 lists prime multiplicities (prime signature), sorted A118914.
%Y A367579 A181819 gives prime shadow, with an inverse A181821.
%Y A367579 A238747 gives prime metasignature, reversed A353742.
%Y A367579 A304038 lists distinct prime indices, length A001221, sum A066328.
%Y A367579 A367582 counts partitions by sum of multiset multiplicity kernel.
%Y A367579 Cf. A000720, A001597, A005117, A027746, A027748, A051904, A052409, A061395, A062770, A175781, A288636, A289023.
%K A367579 nonn,tabf
%O A367579 1,2
%A A367579 _Gus Wiseman_, Nov 25 2023