A362050 Numbers whose prime indices satisfy: (length) = 2*(median).

4, 54, 81, 90, 100, 126, 135, 140, 189, 198, 220, 234, 260, 297, 306, 340, 342, 351, 380, 414, 459, 460, 513, 522, 558, 580, 620, 621, 666, 738, 740, 774, 783, 820, 837, 846, 860, 940, 954, 999, 1060, 1062, 1098, 1107, 1161, 1180, 1206, 1220, 1269, 1278, 1314

Offset: 1

Author

Gus Wiseman, Apr 20 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).

All terms are squarefree.

Examples

			The terms together with their prime indices begin:
    4: {1,1}
   54: {1,2,2,2}
   81: {2,2,2,2}
   90: {1,2,2,3}
  100: {1,1,3,3}
  126: {1,2,2,4}
  135: {2,2,2,3}
  140: {1,1,3,4}
  189: {2,2,2,4}
  198: {1,2,2,5}
		

Crossrefs

The LHS is A001222 (bigomega).

The RHS is A360005 (twice median).

Before multiplying the median by 2, A361800 counts partitions of this type.

For maximum instead of length we have A361856, counted by A361849.

Partitions of this type are counted by A362049.

A061395 gives greatest prime index, least A055396.

A112798 lists prime indices, sum A056239.

A326567/A326568 gives mean of prime indices.

Cf. A067801, A106529, A111907, A255201, A316413, A361868, A361908.

Programs

Mathematica

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