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

1,2

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.

Links

Table of n, a(n) for n=1..27.

Example

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}

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[#]]&]

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.

Sequence in context: A058155 A334527 A104169 * A271615 A271616 A123672

Adjacent sequences: A359901 A359902 A359903 * A359905 A359906 A359907

Keyword

nonn

Author

Gus Wiseman, Jan 25 2023

Status

approved