A360246 Numbers for which the prime indices do not have the same mean as the distinct prime indices.

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189

Offset

1,1

Comments

First differs from A242416 in having 126.

Contains no squarefree numbers or perfect powers.

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.

Links

Table of n, a(n) for n=1..57.

Example

The terms together with their prime indices begin:
   12: {1,1,2}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   28: {1,1,4}
   40: {1,1,1,3}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}
   50: {1,3,3}
   52: {1,1,6}
   54: {1,2,2,2}
   56: {1,1,1,4}
   60: {1,1,2,3}
   63: {2,2,4}
   68: {1,1,7}
   72: {1,1,1,2,2}
The prime indices of 126 are {1,2,2,4} with mean 9/4 and distinct prime indices {1,2,4} with mean 7/3, so 126 is in the sequence.

Mathematica

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

Crossrefs

Signature instead of parts: complement A324570, counted by A114638.

Signature instead of distinct parts: complement A359903, counted by A360068.

These partitions are counted by A360242.

The complement is A360247, counted by A360243.

For median we have A360248, counted by A360244 (complement A360245).

Union of A360252 and A360253, counted by A360250 and A360251.

A058398 counts partitions by mean, also A327482.

A088529/A088530 gives mean of prime signature (A124010).

A112798 lists prime indices, length A001222, sum A056239.

A316413 = numbers whose prime indices have integer mean, distinct A326621.

A326567/A326568 gives mean of prime indices.

A326619/A326620 gives mean of distinct prime indices.

Cf. A000975, A051293, A067340, A067538, A360005, A360241.

Sequence in context: A323055 A059404 A303946 * A242416 A360248 A317616

Adjacent sequences: A360243 A360244 A360245 * A360247 A360248 A360249

Keyword

nonn

Author

Gus Wiseman, Feb 07 2023

Status

approved