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

1, 4, 27, 400, 3125, 9072, 10800, 14580, 24057, 35721, 50625, 73984, 117760, 134400, 158976, 181440, 191488, 389376, 452709, 544000, 583680, 664848, 731136, 774400, 823543, 878592, 965888

Offset: 1

Author

Gus Wiseman, Jan 25 2023

Keywords

• nonn

Comments

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

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

Examples

			The terms together with their prime factors begin:
      1: {}
      4: {2,2}
     27: {3,3,3}
    400: {2,2,2,2,5,5}
   3125: {5,5,5,5,5}
   9072: {2,2,2,2,3,3,3,3,7}
  10800: {2,2,2,2,3,3,3,5,5}
  14580: {2,2,3,3,3,3,3,3,5}
  24057: {3,3,3,3,3,3,3,11}
  35721: {3,3,3,3,3,3,7,7}
  50625: {3,3,3,3,5,5,5,5}
  73984: {2,2,2,2,2,2,2,2,17,17}
		

Crossrefs

The prime factors are A027746, mean A123528/A123529.

The prime signature is A124010, mean A088529/A088530.

For prime indices instead of factors we have A359903.

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

A067340 lists numbers whose prime signature has integer mean.

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

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

A360005 gives median of prime indices (times two).

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

Programs

Mathematica

prifac[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
prisig[n_]:=If[n==1,{},Last/@FactorInteger[n]];
Select[Range[1000],Mean[prifac[#]]==Mean[prisig[#]]&]