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Numbers whose multiset of prime factors (or indices, see A112798) has more adjacent equalities (or parts that have appeared before) than distinct parts.
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%I #6 Feb 21 2023 23:24:39

%S 8,16,27,32,48,64,72,80,81,96,108,112,125,128,144,160,162,176,192,200,

%T 208,216,224,243,256,272,288,304,320,324,343,352,368,384,392,400,405,

%U 416,432,448,464,480,486,496,500,512,544,567,576,592,608,625,640,648

%N Numbers whose multiset of prime factors (or indices, see A112798) has more adjacent equalities (or parts that have appeared before) than distinct parts.

%C No terms are squarefree.

%C Also numbers whose first differences of 0-prepended prime indices have median 0.

%F A001222(a(n)) > 2*A001221(a(n)).

%e The terms together with their prime indices begin:

%e 8: {1,1,1}

%e 16: {1,1,1,1}

%e 27: {2,2,2}

%e 32: {1,1,1,1,1}

%e 48: {1,1,1,1,2}

%e 64: {1,1,1,1,1,1}

%e 72: {1,1,1,2,2}

%e 80: {1,1,1,1,3}

%e 81: {2,2,2,2}

%e 96: {1,1,1,1,1,2}

%e 108: {1,1,2,2,2}

%e 112: {1,1,1,1,4}

%e 125: {3,3,3}

%e For example, the prime indices of 720 are {1,1,1,1,2,2,3} with 4 adjacent equalities and 3 distinct parts, so 720 is in the sequence.

%t Select[Range[100],PrimeOmega[#]>2*PrimeNu[#]&]

%Y For equality we have A067801.

%Y These partitions are counted by A360254.

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

%Y A326567/A326568 gives mean of prime indices.

%Y A360005 gives median of prime indices (times 2).

%Y Cf. A000975, A027193, A067340, A237363, A317090, A360248, A360249, A360454, A360555, A360556.

%K nonn

%O 1,1

%A _Gus Wiseman_, Feb 20 2023