A363949 Numbers whose prime indices have mean 1 when rounded down.

2, 4, 6, 8, 12, 16, 18, 20, 24, 32, 36, 40, 48, 54, 56, 60, 64, 72, 80, 96, 108, 112, 120, 128, 144, 160, 162, 168, 176, 180, 192, 200, 216, 224, 240, 256, 288, 320, 324, 336, 352, 360, 384, 400, 416, 432, 448, 480, 486, 504, 512, 528, 540, 560, 576, 600, 640

Offset: 1

Author

Gus Wiseman, Jul 02 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.

Examples

			The terms together with their prime indices begin:
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   24: {1,1,1,2}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   48: {1,1,1,1,2}
   54: {1,2,2,2}
   56: {1,1,1,4}
   60: {1,1,2,3}
   64: {1,1,1,1,1,1}
		

Crossrefs

These partitions are counted by A025065.

Before rounding down we had A326567/A326568.

For mode instead of mean we have A360015, counted by A241131.

For median instead of mean we have A363488, counted by A027336.

Positions of 1's in A363943, triangle A363945.

For the usual rounding (not low or high) we have A363948, counted by A363947.

A112798 lists prime indices, length A001222, sum A056239.

A316413 ranks partitions with integer mean, counted by A067538.

A360005 gives twice the median of prime indices.

A363941 gives low median of prime indices, triangle A124943.

A363942 gives high median of prime indices, triangle A124944.

For mean 2 instead of 1 we have A363950, counted by A026905 redoubled.

Cf. A051293, A327473, A327476, A344296, A359889, A360013, A363723, A363727, A363946, A363951.

Programs

Mathematica

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

Formula

a(n) = 2*A344296(n).