A359904 - OEIS

%I #7 Jan 26 2023 10:04:32

%S 1,4,27,400,3125,9072,10800,14580,24057,35721,50625,73984,117760,

%T 134400,158976,181440,191488,389376,452709,544000,583680,664848,

%U 731136,774400,823543,878592,965888

%N Numbers whose prime factors and prime signature have the same mean.

%C The multiset of prime factors of n is row n of A027746.

%C A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

%e The terms together with their prime factors begin:

%e       1: {}

%e       4: {2,2}

%e      27: {3,3,3}

%e     400: {2,2,2,2,5,5}

%e    3125: {5,5,5,5,5}

%e    9072: {2,2,2,2,3,3,3,3,7}

%e   10800: {2,2,2,2,3,3,3,5,5}

%e   14580: {2,2,3,3,3,3,3,3,5}

%e   24057: {3,3,3,3,3,3,3,11}

%e   35721: {3,3,3,3,3,3,7,7}

%e   50625: {3,3,3,3,5,5,5,5}

%e   73984: {2,2,2,2,2,2,2,2,17,17}

%t prifac[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];

%t prisig[n_]:=If[n==1,{},Last/@FactorInteger[n]];

%t Select[Range[1000],Mean[prifac[#]]==Mean[prisig[#]]&]

%Y The prime factors are A027746, mean A123528/A123529.

%Y The prime signature is A124010, mean A088529/A088530.

%Y For prime indices instead of factors we have A359903.

%Y A058398 counts partitions by mean, see also A008284, A327482.

%Y A067340 lists numbers whose prime signature has integer mean.

%Y A078175 = numbers whose prime factors have integer mean, indices A316413.

%Y A112798 = prime indices, length A001222, sum A056239, mean A326567/A326568.

%Y A360005 gives median of prime indices (times two).

%Y Cf. A240219, A326622, A327473, A359905, A359908, A359913, A360008, A360068.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jan 25 2023