A363944 Mean of the multiset of prime indices of n, rounded up.

0, 1, 2, 1, 3, 2, 4, 1, 2, 2, 5, 2, 6, 3, 3, 1, 7, 2, 8, 2, 3, 3, 9, 2, 3, 4, 2, 2, 10, 2, 11, 1, 4, 4, 4, 2, 12, 5, 4, 2, 13, 3, 14, 3, 3, 5, 15, 2, 4, 3, 5, 3, 16, 2, 4, 2, 5, 6, 17, 2, 18, 6, 3, 1, 5, 3, 19, 3, 6, 3, 20, 2, 21, 7, 3, 4, 5, 3, 22, 2, 2, 7

Offset: 1

Author

Gus Wiseman, Jun 30 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.

Extending the terminology introduced at A124944, this is the "high mean" of prime indices.

Examples

			The prime indices of 360 are {1,1,1,2,2,3}, with mean 3/2, so a(360) = 2.
		

Crossrefs

Positions of first appearances are 1 and A000040.

Positions of 1's are A000079(n>0).

Before rounding up we had A326567/A326568.

For mode instead of mean we have A363487, low A363486.

For median instead of mean we have A363942, triangle A124944.

Rounding down instead of up gives A363943, triangle A363945.

The triangle for this statistic (high mean) is A363946.

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.

A363947 ranks partitions with rounded mean 1, counted by A363948.

A363949 ranks partitions with low mean 1, counted by A025065.

A363950 ranks partitions with low mean 2, counted by A026905 redoubled.

Cf. A051293, A124943, A215366, A327473, A327476, A327482, A359889, A362611, A363723, A363724, A363727, A363951.

Programs

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
meanup[y_]:=If[Length[y]==0,0,Ceiling[Mean[y]]];
Table[meanup[prix[n]],{n,100}]