cp's OEIS Frontend

A340387 Numbers whose sum of prime indices is twice their number, counted with multiplicity in both cases.

%I A340387 #15 Jan 15 2021 09:16:02
%S A340387 1,3,9,10,27,28,30,81,84,88,90,100,208,243,252,264,270,280,300,544,
%T A340387 624,729,756,784,792,810,840,880,900,1000,1216,1632,1872,2080,2187,
%U A340387 2268,2352,2376,2430,2464,2520,2640,2700,2800,2944,3000,3648,4896,5440,5616
%N A340387 Numbers whose sum of prime indices is twice their number, counted with multiplicity in both cases.
%C A340387 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 A340387 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 A340387 All terms satisfy A056239(a(n)) = 2*A001222(a(n)).
%e A340387 The sequence of terms together with their prime indices begins:
%e A340387       1: {}
%e A340387       3: {2}
%e A340387       9: {2,2}
%e A340387      10: {1,3}
%e A340387      27: {2,2,2}
%e A340387      28: {1,1,4}
%e A340387      30: {1,2,3}
%e A340387      81: {2,2,2,2}
%e A340387      84: {1,1,2,4}
%e A340387      88: {1,1,1,5}
%e A340387      90: {1,2,2,3}
%e A340387     100: {1,1,3,3}
%e A340387     208: {1,1,1,1,6}
%e A340387     243: {2,2,2,2,2}
%e A340387     252: {1,1,2,2,4}
%t A340387 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A340387 Select[Range[1000],Total[primeMS[#]]==2*PrimeOmega[#]&]
%Y A340387 Partitions of 2n into n parts are counted by A000041.
%Y A340387 The number of prime indices alone is A001222.
%Y A340387 The sum of prime indices alone is A056239.
%Y A340387 Allowing sum to be any multiple of length gives A067538, ranked by A316413.
%Y A340387 A000569 counts graphical partitions, ranked by A320922.
%Y A340387 A027187 counts partitions of even length, ranked by A028260.
%Y A340387 A058696 counts partitions of even numbers, ranked by A300061.
%Y A340387 A301987 lists numbers whose sum of prime indices equals their product, with nonprime case A301988.
%Y A340387 Cf. A000720, A001221, A001414, A006125, A006129, A112798, A316428, A320911, A325037, A325044, A330950, A331385, A331416.
%K A340387 nonn
%O A340387 1,2
%A A340387 _Gus Wiseman_, Jan 09 2021