login
Irregular triangle read by rows where row n is the multiset multiplicity kernel (MMK) of the multiset of prime indices of n.
15

%I #16 Nov 27 2023 07:57:30

%S 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 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 5,2,3,1,1,15,1,2,4,1,3,2,2,1,6,16,1,2

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

%C Row n = 1 is empty.

%C 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 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 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 For all positive integers n and k, row n^k is the same as row n.

%e The first 45 rows:

%e 1: {} 16: {1} 31: {11}

%e 2: {1} 17: {7} 32: {1}

%e 3: {2} 18: {1,2} 33: {2,2}

%e 4: {1} 19: {8} 34: {1,1}

%e 5: {3} 20: {1,3} 35: {3,3}

%e 6: {1,1} 21: {2,2} 36: {1,1}

%e 7: {4} 22: {1,1} 37: {12}

%e 8: {1} 23: {9} 38: {1,1}

%e 9: {2} 24: {1,2} 39: {2,2}

%e 10: {1,1} 25: {3} 40: {1,3}

%e 11: {5} 26: {1,1} 41: {13}

%e 12: {1,2} 27: {2} 42: {1,1,1}

%e 13: {6} 28: {1,4} 43: {14}

%e 14: {1,1} 29: {10} 44: {1,5}

%e 15: {2,2} 30: {1,1,1} 45: {2,3}

%t mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q,Count[q,#]==i&], {i,mts}]]];

%t Table[mmk[PrimePi/@Join@@ConstantArray@@@If[n==1, {},FactorInteger[n]]], {n,100}]

%Y Indices of empty and singleton rows are A000961.

%Y Row lengths are A001221.

%Y Depends only on rootless base A052410, see A007916.

%Y Row minima are A055396.

%Y Rows have A071625 distinct elements.

%Y Indices of constant rows are A072774.

%Y Indices of strict rows are A130091.

%Y Rows have Heinz numbers A367580.

%Y Row sums are A367581.

%Y Row maxima are A367583, opposite A367587.

%Y Index of first row with Heinz number n is A367584.

%Y Sorted row indices of first appearances are A367585.

%Y Indices of rows of the form {1,1,...} are A367586.

%Y Agrees with sorted prime signature at A367683, counted by A367682.

%Y A submultiset of prime indices at A367685, counted by A367684.

%Y A007947 gives squarefree kernel.

%Y A112798 lists prime indices, length A001222, sum A056239, reverse A296150.

%Y A124010 lists prime multiplicities (prime signature), sorted A118914.

%Y A181819 gives prime shadow, with an inverse A181821.

%Y A238747 gives prime metasignature, reversed A353742.

%Y A304038 lists distinct prime indices, length A001221, sum A066328.

%Y A367582 counts partitions by sum of multiset multiplicity kernel.

%Y Cf. A000720, A001597, A005117, A027746, A027748, A051904, A052409, A061395, A062770, A175781, A288636, A289023.

%K nonn,tabf

%O 1,2

%A _Gus Wiseman_, Nov 25 2023