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A359903 Numbers whose prime indices and prime signature have the same mean.

%I A359903 #7 Jan 26 2023 10:04:23
%S A359903 1,2,9,88,100,125,624,756,792,810,880,900,1312,2401,4617,4624,6240,
%T A359903 7392,7560,7920,8400,9261,9604,9801,10648,12416,23424,33984,37760,
%U A359903 45792,47488,60912,66176,71552,73920,75200,78720,83592,89216,89984,91264,91648,99456
%N A359903 Numbers whose prime indices and prime signature have the same mean.
%C A359903 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 A359903 A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.
%e A359903 The terms together with their prime indices begin:
%e A359903       1: {}
%e A359903       2: {1}
%e A359903       9: {2,2}
%e A359903      88: {1,1,1,5}
%e A359903     100: {1,1,3,3}
%e A359903     125: {3,3,3}
%e A359903     624: {1,1,1,1,2,6}
%e A359903     756: {1,1,2,2,2,4}
%e A359903     792: {1,1,1,2,2,5}
%e A359903     810: {1,2,2,2,2,3}
%e A359903     880: {1,1,1,1,3,5}
%e A359903     900: {1,1,2,2,3,3}
%e A359903    1312: {1,1,1,1,1,13}
%e A359903    2401: {4,4,4,4}
%e A359903    4617: {2,2,2,2,2,8}
%e A359903    4624: {1,1,1,1,7,7}
%e A359903    6240: {1,1,1,1,1,2,3,6}
%e A359903    7392: {1,1,1,1,1,2,4,5}
%e A359903    7560: {1,1,1,2,2,2,3,4}
%e A359903    7920: {1,1,1,1,2,2,3,5}
%e A359903 Example: 810 has prime indices {1,2,2,2,2,3} and prime exponents (1,4,1), both of which have mean 2, so 810 is in the sequence.
%e A359903 Example: 78720 has prime indices {1,1,1,1,1,1,1,2,3,13} and prime exponents (7,1,1,1), both of which have mean 5/2, so 78720 is in the sequence.
%t A359903 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A359903 prisig[n_]:=If[n==1,{},Last/@FactorInteger[n]];
%t A359903 Select[Range[1000],Mean[prix[#]]==Mean[prisig[#]]&]
%Y A359903 Prime indices are A112798, sum A056239, mean A326567/A326568.
%Y A359903 Prime signature is A124010, sum A001222, mean A088529/A088530.
%Y A359903 For prime factors instead of indices we have A359904.
%Y A359903 Partitions with these Heinz numbers are counted by A360068.
%Y A359903 A058398 counts partitions by mean, see also A008284, A327482.
%Y A359903 A067340 lists numbers whose prime signature has integer mean.
%Y A359903 A316413 lists numbers whose prime indices have integer mean.
%Y A359903 A360005 gives median of prime indices (times two).
%Y A359903 Cf. A240219, A316313, A326622, A327473, A327476, A348551, A359905, A359908, A360008, A360069.
%K A359903 nonn
%O A359903 1,2
%A A359903 _Gus Wiseman_, Jan 24 2023