A359913 Numbers whose multiset of prime factors has integer median.

2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 35, 37, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81

Offset: 1

Author

Gus Wiseman, Jan 25 2023

Keywords

• nonn

Comments

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Examples

			The terms together with their prime factors begin:
   2: {2}
   3: {3}
   4: {2,2}
   5: {5}
   7: {7}
   8: {2,2,2}
   9: {3,3}
  11: {11}
  12: {2,2,3}
  13: {13}
  15: {3,5}
  16: {2,2,2,2}
  17: {17}
  18: {2,3,3}
  19: {19}
  20: {2,2,5}
  21: {3,7}
  23: {23}
  24: {2,2,2,3}
		

Crossrefs

Prime factors are listed by A027746.

The complement is A072978, for prime indices A359912.

For mean instead of median we have A078175, for prime indices A316413.

For prime indices instead of factors we have A359908, counted by A325347.

Positions of even terms in A360005.

A067340 lists numbers whose prime signature has integer mean.

A112798 lists prime indices, length A001222, sum A056239.

A325347 counts partitions with integer median, strict A359907.

A326567/A326568 gives the mean of prime indices, conjugate A326839/A326840.

A359893 and A359901 count partitions by median, odd-length A359902.

Cf. A067538, A175352, A348551, A359890, A359905, A360007, A360006, A360069.

Programs

Mathematica

Select[Range[2,100],IntegerQ[Median[Flatten[ConstantArray@@@FactorInteger[#]]]]&]