A363489 Rounded mean of the multiset of prime indices of n.

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

Offset: 1

Author

Gus Wiseman, Jul 07 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.

We use the "rounding half to even" rule, see link.

Examples

			The prime indices of 180 are {1,1,2,2,3}, with mean 9/5, which rounds to 2, so a(180) = 2.
		

Links

• Wikipedia, Rounding.

Crossrefs

Positions of first appearances are 1 and A000040.

Before rounding we had A326567/A326568.

For rounded-down: A363943, triangle A363945.

For rounded-up: A363944, triangle A363946.

Positions of 1's are A363948, complement A364059.

The triangle for this statistic (rounded mean) is A364060.

For prime factors instead of indices we have A364061.

A088529/A088530 gives mean of prime signature A124010.

A112798 lists prime indices, length A001222, sum A056239.

A316413 ranks partitions with integer mean, counted by A067538.

Cf. A327473, A327476, A327482, A359889, A360005, A363947, A363949, A363951.

Programs

Mathematica

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