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A363218 Positive integers whose prime indices satisfy: (length) = 2*(maximum).

%I A363218 #7 May 27 2023 16:46:01
%S A363218 4,24,36,54,81,160,240,360,400,540,600,810,896,900,1000,1215,1344,
%T A363218 1350,1500,2016,2025,2240,2250,2500,3024,3136,3360,3375,3750,4536,
%U A363218 4704,5040,5600,5625,5632,6250,6804,7056,7560,7840,8400,8448,9375,10206,10584,10976
%N A363218 Positive integers whose prime indices satisfy: (length) = 2*(maximum).
%C A363218 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.
%F A363218 Disjoint from A361909.
%e A363218 The terms together with their prime indices begin:
%e A363218       4: {1,1}
%e A363218      24: {1,1,1,2}
%e A363218      36: {1,1,2,2}
%e A363218      54: {1,2,2,2}
%e A363218      81: {2,2,2,2}
%e A363218     160: {1,1,1,1,1,3}
%e A363218     240: {1,1,1,1,2,3}
%e A363218     360: {1,1,1,2,2,3}
%e A363218     400: {1,1,1,1,3,3}
%e A363218     540: {1,1,2,2,2,3}
%e A363218     600: {1,1,1,2,3,3}
%e A363218     810: {1,2,2,2,2,3}
%e A363218     896: {1,1,1,1,1,1,1,4}
%e A363218     900: {1,1,2,2,3,3}
%e A363218    1000: {1,1,1,3,3,3}
%e A363218    1215: {2,2,2,2,2,3}
%e A363218    1344: {1,1,1,1,1,1,2,4}
%e A363218    1350: {1,2,2,2,3,3}
%e A363218    1500: {1,1,2,3,3,3}
%e A363218    2016: {1,1,1,1,1,2,2,4}
%e A363218    2025: {2,2,2,2,3,3}
%e A363218    2240: {1,1,1,1,1,1,3,4}
%t A363218 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A363218 Select[Range[1000],Length[prix[#]]==2*Max[prix[#]]&]
%Y A363218 The LHS (number of prime indices) is A001222.
%Y A363218 The RHS is twice A061395.
%Y A363218 Before multiplying by 2 we had A106529.
%Y A363218 Partitions of this type are counted by A237753.
%Y A363218 For sum instead of length we have A344415, counted by A035363.
%Y A363218 An adjoint version is A361909, also counted by A237753.
%Y A363218 For minimum instead of maximum we have A363134, counted by A237757.
%Y A363218 A112798 lists prime indices, sum A056239.
%Y A363218 A326567/A326568 gives mean of prime indices.
%Y A363218 Cf. A001221, A067801, A118096, A324521, A237752, A361855, A361908, A363222.
%K A363218 nonn
%O A363218 1,1
%A A363218 _Gus Wiseman_, May 23 2023