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Numbers whose sum of prime indices is twice their number, counted with multiplicity in both cases.
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%I #15 Jan 15 2021 09:16:02

%S 1,3,9,10,27,28,30,81,84,88,90,100,208,243,252,264,270,280,300,544,

%T 624,729,756,784,792,810,840,880,900,1000,1216,1632,1872,2080,2187,

%U 2268,2352,2376,2430,2464,2520,2640,2700,2800,2944,3000,3648,4896,5440,5616

%N Numbers whose sum of prime indices is twice their number, counted with multiplicity in both cases.

%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 Also Heinz numbers of integer partitions whose sum is twice their length, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). Like partitions in general (A000041), these are also counted by A000041.

%F All terms satisfy A056239(a(n)) = 2*A001222(a(n)).

%e The sequence of terms together with their prime indices begins:

%e 1: {}

%e 3: {2}

%e 9: {2,2}

%e 10: {1,3}

%e 27: {2,2,2}

%e 28: {1,1,4}

%e 30: {1,2,3}

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

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

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

%e 90: {1,2,2,3}

%e 100: {1,1,3,3}

%e 208: {1,1,1,1,6}

%e 243: {2,2,2,2,2}

%e 252: {1,1,2,2,4}

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

%t Select[Range[1000],Total[primeMS[#]]==2*PrimeOmega[#]&]

%Y Partitions of 2n into n parts are counted by A000041.

%Y The number of prime indices alone is A001222.

%Y The sum of prime indices alone is A056239.

%Y Allowing sum to be any multiple of length gives A067538, ranked by A316413.

%Y A000569 counts graphical partitions, ranked by A320922.

%Y A027187 counts partitions of even length, ranked by A028260.

%Y A058696 counts partitions of even numbers, ranked by A300061.

%Y A301987 lists numbers whose sum of prime indices equals their product, with nonprime case A301988.

%Y Cf. A000720, A001221, A001414, A006125, A006129, A112798, A316428, A320911, A325037, A325044, A330950, A331385, A331416.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jan 09 2021