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%I #6 Apr 20 2023 14:56:08
%S 4,54,81,90,100,126,135,140,189,198,220,234,260,297,306,340,342,351,
%T 380,414,459,460,513,522,558,580,620,621,666,738,740,774,783,820,837,
%U 846,860,940,954,999,1060,1062,1098,1107,1161,1180,1206,1220,1269,1278,1314
%N Numbers whose prime indices satisfy: (length) = 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).
%C All terms are squarefree.
%e The terms together with their prime indices begin:
%e 4: {1,1}
%e 54: {1,2,2,2}
%e 81: {2,2,2,2}
%e 90: {1,2,2,3}
%e 100: {1,1,3,3}
%e 126: {1,2,2,4}
%e 135: {2,2,2,3}
%e 140: {1,1,3,4}
%e 189: {2,2,2,4}
%e 198: {1,2,2,5}
%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Select[Range[100],PrimeOmega[#]==2*Median[prix[#]]&]
%Y The LHS is A001222 (bigomega).
%Y The RHS is A360005 (twice median).
%Y Before multiplying the median by 2, A361800 counts partitions of this type.
%Y For maximum instead of length we have A361856, counted by A361849.
%Y Partitions of this type are counted by A362049.
%Y A061395 gives greatest prime index, least A055396.
%Y A112798 lists prime indices, sum A056239.
%Y A326567/A326568 gives mean of prime indices.
%Y Cf. A067801, A106529, A111907, A255201, A316413, A361868, A361908.
%K nonn
%O 1,1
%A _Gus Wiseman_, Apr 20 2023
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