%I #6 Apr 06 2023 21:44:06

%S 20,28,40,44,52,56,66,68,76,78,80,84,88,92,99,102,104,112,114,116,117,

%T 120,124,132,136,138,148,152,153,156,160,164,168,170,171,172,174,176,

%U 184,186,188,190,198,200,204,207,208,212,220,222,224,228,230,232,234

%N Positive integers > 1 whose prime indices satisfy (maximum) > 2*(median).

%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 The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

%e The prime indices of 84 are {1,1,2,4}, with maximum 4 and median 3/2, and 4 > 2*(3/2), so 84 is in the sequence.

%e The terms together with their prime indices begin:

%e 20: {1,1,3}

%e 28: {1,1,4}

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

%e 44: {1,1,5}

%e 52: {1,1,6}

%e 56: {1,1,1,4}

%e 66: {1,2,5}

%e 68: {1,1,7}

%e 76: {1,1,8}

%e 78: {1,2,6}

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

%e 84: {1,1,2,4}

%e 88: {1,1,1,5}

%e 92: {1,1,9}

%e 99: {2,2,5}

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100], Max@@prix[#]>2*Median[prix[#]]&]

%Y The LHS is A061395 (greatest prime index).

%Y The RHS is A360005 (twice median), distinct A360457.

%Y The equal version is A361856, counted by A361849.

%Y These partitions are counted by A361857, reverse A361858.

%Y Including the equal case gives A361868, counted by A361859.

%Y For mean instead of median we have A361907.

%Y A000975 counts subsets with integer median.

%Y A001222 counts prime factors, distinct A001221.

%Y A112798 lists prime indices, sum A056239.

%Y Cf. A053263, A067801, A111907, A237751, A237820, A359893, A361848, A361855, A361908, A361909.

%K nonn

%O 1,1

%A _Gus Wiseman_, Apr 05 2023