cp's OEIS Frontend

A359904 Numbers whose prime factors and prime signature have the same mean.

%I A359904 #7 Jan 26 2023 10:04:32
%S A359904 1,4,27,400,3125,9072,10800,14580,24057,35721,50625,73984,117760,
%T A359904 134400,158976,181440,191488,389376,452709,544000,583680,664848,
%U A359904 731136,774400,823543,878592,965888
%N A359904 Numbers whose prime factors and prime signature have the same mean.
%C A359904 The multiset of prime factors of n is row n of A027746.
%C A359904 A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.
%e A359904 The terms together with their prime factors begin:
%e A359904       1: {}
%e A359904       4: {2,2}
%e A359904      27: {3,3,3}
%e A359904     400: {2,2,2,2,5,5}
%e A359904    3125: {5,5,5,5,5}
%e A359904    9072: {2,2,2,2,3,3,3,3,7}
%e A359904   10800: {2,2,2,2,3,3,3,5,5}
%e A359904   14580: {2,2,3,3,3,3,3,3,5}
%e A359904   24057: {3,3,3,3,3,3,3,11}
%e A359904   35721: {3,3,3,3,3,3,7,7}
%e A359904   50625: {3,3,3,3,5,5,5,5}
%e A359904   73984: {2,2,2,2,2,2,2,2,17,17}
%t A359904 prifac[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
%t A359904 prisig[n_]:=If[n==1,{},Last/@FactorInteger[n]];
%t A359904 Select[Range[1000],Mean[prifac[#]]==Mean[prisig[#]]&]
%Y A359904 The prime factors are A027746, mean A123528/A123529.
%Y A359904 The prime signature is A124010, mean A088529/A088530.
%Y A359904 For prime indices instead of factors we have A359903.
%Y A359904 A058398 counts partitions by mean, see also A008284, A327482.
%Y A359904 A067340 lists numbers whose prime signature has integer mean.
%Y A359904 A078175 = numbers whose prime factors have integer mean, indices A316413.
%Y A359904 A112798 = prime indices, length A001222, sum A056239, mean A326567/A326568.
%Y A359904 A360005 gives median of prime indices (times two).
%Y A359904 Cf. A240219, A326622, A327473, A359905, A359908, A359913, A360008, A360068.
%K A359904 nonn
%O A359904 1,2
%A A359904 _Gus Wiseman_, Jan 25 2023