A361868 Positive integers > 1 whose prime indices satisfy (maximum) >= 2*(median).

12, 20, 24, 28, 40, 42, 44, 48, 52, 56, 60, 63, 66, 68, 72, 76, 78, 80, 84, 88, 92, 96, 99, 102, 104, 112, 114, 116, 117, 120, 124, 126, 130, 132, 136, 138, 140, 144, 148, 152, 153, 156, 160, 164, 168, 170, 171, 172, 174, 176, 184, 186, 188, 189, 190, 192, 195

Offset: 1

Author

Gus Wiseman, Apr 05 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 prime indices of 84 are {1,1,2,4}, with maximum 4 and median 3/2, and 4 >= 2*(3/2), so 84 is in the sequence.
The terms together with their prime indices begin:
   12: {1,1,2}
   20: {1,1,3}
   24: {1,1,1,2}
   28: {1,1,4}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   48: {1,1,1,1,2}
   52: {1,1,6}
   56: {1,1,1,4}
   60: {1,1,2,3}
   63: {2,2,4}
   66: {1,2,5}
   68: {1,1,7}
   72: {1,1,1,2,2}
		

Crossrefs

The LHS is A061395 (greatest prime index).

The RHS is A360005 (twice median), distinct A360457.

The equal case is A361856, counted by A361849.

These partitions are counted by A361859.

The unequal case is A361867, counted by A361857.

The complement is counted by A361858.

A000975 counts subsets with integer median.

A001222 (bigomega) counts prime factors, distinct A001221 (omega).

A112798 lists prime indices, sum A056239.

A325347 counts partitions with integer median, complement A307683.

A359893 and A359901 count partitions by median.

Cf. A027193, A053263, A111907, A118096, A237753, A237821, A361848, A361855, A361906, A361908.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Max@@prix[#]>=2*Median[prix[#]]&]