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Positive integers whose prime indices satisfy: (length) = 2*(maximum).
3

%I #7 May 27 2023 16:46:01

%S 4,24,36,54,81,160,240,360,400,540,600,810,896,900,1000,1215,1344,

%T 1350,1500,2016,2025,2240,2250,2500,3024,3136,3360,3375,3750,4536,

%U 4704,5040,5600,5625,5632,6250,6804,7056,7560,7840,8400,8448,9375,10206,10584,10976

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

%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.

%F Disjoint from A361909.

%e The terms together with their prime indices begin:

%e 4: {1,1}

%e 24: {1,1,1,2}

%e 36: {1,1,2,2}

%e 54: {1,2,2,2}

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

%e 160: {1,1,1,1,1,3}

%e 240: {1,1,1,1,2,3}

%e 360: {1,1,1,2,2,3}

%e 400: {1,1,1,1,3,3}

%e 540: {1,1,2,2,2,3}

%e 600: {1,1,1,2,3,3}

%e 810: {1,2,2,2,2,3}

%e 896: {1,1,1,1,1,1,1,4}

%e 900: {1,1,2,2,3,3}

%e 1000: {1,1,1,3,3,3}

%e 1215: {2,2,2,2,2,3}

%e 1344: {1,1,1,1,1,1,2,4}

%e 1350: {1,2,2,2,3,3}

%e 1500: {1,1,2,3,3,3}

%e 2016: {1,1,1,1,1,2,2,4}

%e 2025: {2,2,2,2,3,3}

%e 2240: {1,1,1,1,1,1,3,4}

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

%t Select[Range[1000],Length[prix[#]]==2*Max[prix[#]]&]

%Y The LHS (number of prime indices) is A001222.

%Y The RHS is twice A061395.

%Y Before multiplying by 2 we had A106529.

%Y Partitions of this type are counted by A237753.

%Y For sum instead of length we have A344415, counted by A035363.

%Y An adjoint version is A361909, also counted by A237753.

%Y For minimum instead of maximum we have A363134, counted by A237757.

%Y A112798 lists prime indices, sum A056239.

%Y A326567/A326568 gives mean of prime indices.

%Y Cf. A001221, A067801, A118096, A324521, A237752, A361855, A361908, A363222.

%K nonn

%O 1,1

%A _Gus Wiseman_, May 23 2023