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

1,1

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.

Links

Table of n, a(n) for n=1..51.

Example

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}

Mathematica

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

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.

Sequence in context: A307172 A275801 A158259 * A095210 A156469 A001545

Adjacent sequences: A362047 A362048 A362049 * A362051 A362052 A362053

Keyword

nonn

Author

Gus Wiseman, Apr 20 2023

Status

approved