A360009 Numbers whose prime indices have integer mean and integer median.

2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 21, 22, 23, 25, 27, 28, 29, 30, 31, 32, 34, 37, 39, 41, 43, 46, 47, 49, 53, 55, 57, 59, 61, 62, 64, 67, 68, 71, 73, 78, 79, 81, 82, 83, 85, 87, 88, 89, 90, 91, 94, 97, 98, 99, 100, 101, 103, 105, 107, 109, 110, 111

Offset: 1

Author

Gus Wiseman, Jan 24 2023

Keywords

• nonn

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

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 indices begin:
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   21: {2,4}
   22: {1,5}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   28: {1,1,4}
		

Crossrefs

For just integer mean we have A316413 (counted by A067538).

The mean of prime indices is given by A326567/A326568.

The complement is A348551 \/ A359912 (counted by A349156 and A307683).

These partitions are counted by A359906.

For just integer median we have A359908 (counted by A325347).

The median of prime indices is given by A360005/2.

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

A112798 lists prime indices, length A001222, sum A056239.

A326622 counts factorizations with integer mean, strict A328966.

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

Cf. A026424, A327473, A359889, A359890, A359903, A359905, A360006.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],IntegerQ[Mean[prix[#]]]&&IntegerQ[Median[prix[#]]]&]

Formula

Intersection of A316413 and A359908.