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Numbers in whose prime factorization the greatest factor is the unique mode.
20

%I #8 May 07 2023 08:38:42

%S 2,3,4,5,7,8,9,11,13,16,17,18,19,23,25,27,29,31,32,37,41,43,47,49,50,

%T 53,54,59,61,64,67,71,73,75,79,81,83,89,97,98,101,103,107,108,109,113,

%U 121,125,127,128,131,137,139,147,149,150,151,157,162,163,167

%N Numbers in whose prime factorization the greatest factor is the unique mode.

%C First differs from A329131 in lacking 450 and having 1500.

%C A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.

%e The factorization of 90 is 2*3*3*5, modes {3}, so 90 is missing.

%e The factorization of 450 is 2*3*3*5*5, modes {3,5}, so 450 is missing.

%e The factorization of 900 is 2*2*3*3*5*5, modes {2,3,5}, so 900 is missing.

%e The factorization of 1500 is 2*2*3*5*5*5, modes {5}, so 1500 is present.

%e The terms together with their prime indices begin:

%e 2: {1} 27: {2,2,2} 67: {19}

%e 3: {2} 29: {10} 71: {20}

%e 4: {1,1} 31: {11} 73: {21}

%e 5: {3} 32: {1,1,1,1,1} 75: {2,3,3}

%e 7: {4} 37: {12} 79: {22}

%e 8: {1,1,1} 41: {13} 81: {2,2,2,2}

%e 9: {2,2} 43: {14} 83: {23}

%e 11: {5} 47: {15} 89: {24}

%e 13: {6} 49: {4,4} 97: {25}

%e 16: {1,1,1,1} 50: {1,3,3} 98: {1,4,4}

%e 17: {7} 53: {16} 101: {26}

%e 18: {1,2,2} 54: {1,2,2,2} 103: {27}

%e 19: {8} 59: {17} 107: {28}

%e 23: {9} 61: {18} 108: {1,1,2,2,2}

%e 25: {3,3} 64: {1,1,1,1,1,1} 109: {29}

%t prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];

%t Select[Range[100],Commonest[prifacs[#]]=={Max[prifacs[#]]}&]

%Y First term with given bigomega is A000079.

%Y For median instead of mode we have A053263.

%Y Partitions of this type are counted by A362612.

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

%Y A356862 ranks partitions with a unique mode, counted by A362608.

%Y A359178 ranks partitions with a unique co-mode, counted by A362610.

%Y A362605 ranks partitions with more than one mode, counted by A362607.

%Y A362606 ranks partitions with more than one co-mode, counted by A362609.

%Y A362614 counts partitions by number of modes, ranked by A362611.

%Y A362615 counts partitions by number of co-modes, ranked by A362613.

%Y Cf. A000040, A002865, A327473, A327476, A358137, A360687.

%K nonn

%O 1,1

%A _Gus Wiseman_, May 05 2023