A367858 - OEIS

%I #12 Jan 19 2024 04:33:43

%S 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 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 5,2,3,9,9,15,1,2,4,1,3,7,7,1,6,16,1,2

%N Irregular triangle read by rows where row n is the multiset multiplicity cokernel (MMC) 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 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 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: {5,5}

%e      4: {1}      19: {8}        34: {7,7}

%e      5: {3}      20: {1,3}      35: {4,4}

%e      6: {2,2}    21: {4,4}      36: {2,2}

%e      7: {4}      22: {5,5}      37: {12}

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

%e      9: {2}      24: {1,2}      39: {6,6}

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

%e     11: {5}      26: {6,6}      41: {13}

%e     12: {1,2}    27: {2}        42: {4,4,4}

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

%e     14: {4,4}    29: {10}       44: {1,5}

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

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

%t Table[mmc[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 maxima are A061395.

%Y Rows have A071625 distinct elements.

%Y Indices of constant rows are A072774.

%Y Indices of strict rows are A130091.

%Y Row minima are A367587.

%Y Rows have Heinz numbers A367859.

%Y Row sums are A367860.

%Y Sorted row indices of first appearances are A367861, for kernel A367585.

%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 Cf. A000720, A027746, A051904, A052409, A061395, A175781, A367582, A367583.

%K nonn,tabf

%O 1,2

%A _Gus Wiseman_, Dec 03 2023