cp's OEIS Frontend

A367858 Irregular triangle read by rows where row n is the multiset multiplicity cokernel (MMC) of the multiset of prime indices of n.

%I A367858 #12 Jan 19 2024 04:33:43
%S A367858 1,2,1,3,2,2,4,1,2,3,3,5,1,2,6,4,4,3,3,1,7,1,2,8,1,3,4,4,5,5,9,1,2,3,
%T A367858 6,6,2,1,4,10,3,3,3,11,1,5,5,7,7,4,4,2,2,12,8,8,6,6,1,3,13,4,4,4,14,1,
%U A367858 5,2,3,9,9,15,1,2,4,1,3,7,7,1,6,16,1,2
%N A367858 Irregular triangle read by rows where row n is the multiset multiplicity cokernel (MMC) of the multiset of prime indices of n.
%C A367858 Row n = 1 is empty.
%C A367858 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 A367858 We define the multiset multiplicity cokernel MMC(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 max(S) has multiplicity |S| in MMC(m). For example, MMC({1,1,2,2,3,4,5}) = {2,2,5,5,5}, and MMC({1,2,3,4,5,5,5,5}) = {4,4,4,4,5}.
%F A367858 For all positive integers n and k, row n^k is the same as row n.
%e A367858 The first 45 rows:
%e A367858      1: {}       16: {1}        31: {11}
%e A367858      2: {1}      17: {7}        32: {1}
%e A367858      3: {2}      18: {1,2}      33: {5,5}
%e A367858      4: {1}      19: {8}        34: {7,7}
%e A367858      5: {3}      20: {1,3}      35: {4,4}
%e A367858      6: {2,2}    21: {4,4}      36: {2,2}
%e A367858      7: {4}      22: {5,5}      37: {12}
%e A367858      8: {1}      23: {9}        38: {8,8}
%e A367858      9: {2}      24: {1,2}      39: {6,6}
%e A367858     10: {3,3}    25: {3}        40: {1,3}
%e A367858     11: {5}      26: {6,6}      41: {13}
%e A367858     12: {1,2}    27: {2}        42: {4,4,4}
%e A367858     13: {6}      28: {1,4}      43: {14}
%e A367858     14: {4,4}    29: {10}       44: {1,5}
%e A367858     15: {3,3}    30: {3,3,3}    45: {2,3}
%t A367858 mmc[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Max@@Select[q,Count[q,#]==i&], {i,mts}]]];
%t A367858 Table[mmc[PrimePi /@ Join@@ConstantArray@@@If[n==1, {},FactorInteger[n]]], {n,100}]
%Y A367858 Indices of empty and singleton rows are A000961.
%Y A367858 Row lengths are A001221.
%Y A367858 Depends only on rootless base A052410, see A007916.
%Y A367858 Row maxima are A061395.
%Y A367858 Rows have A071625 distinct elements.
%Y A367858 Indices of constant rows are A072774.
%Y A367858 Indices of strict rows are A130091.
%Y A367858 Row minima are A367587.
%Y A367858 Rows have Heinz numbers A367859.
%Y A367858 Row sums are A367860.
%Y A367858 Sorted row indices of first appearances are A367861, for kernel A367585.
%Y A367858 A007947 gives squarefree kernel.
%Y A367858 A112798 lists prime indices, length A001222, sum A056239, reverse A296150.
%Y A367858 A124010 lists prime multiplicities (prime signature), sorted A118914.
%Y A367858 A181819 gives prime shadow, with an inverse A181821.
%Y A367858 A238747 gives prime metasignature, reversed A353742.
%Y A367858 A304038 lists distinct prime indices, length A001221, sum A066328.
%Y A367858 Cf. A000720, A027746, A051904, A052409, A061395, A175781, A367582, A367583.
%K A367858 nonn,tabf
%O A367858 1,2
%A A367858 _Gus Wiseman_, Dec 03 2023