A360677 Sum of the right half (exclusive) of the prime indices of n.

0, 0, 0, 1, 0, 2, 0, 1, 2, 3, 0, 2, 0, 4, 3, 2, 0, 2, 0, 3, 4, 5, 0, 3, 3, 6, 2, 4, 0, 3, 0, 2, 5, 7, 4, 4, 0, 8, 6, 4, 0, 4, 0, 5, 3, 9, 0, 3, 4, 3, 7, 6, 0, 4, 5, 5, 8, 10, 0, 5, 0, 11, 4, 3, 6, 5, 0, 7, 9, 4, 0, 4, 0, 12, 3, 8, 5, 6, 0, 4, 4, 13, 0, 6, 7

Offset: 1

Author

Gus Wiseman, Mar 05 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 prime indices of 810 are {1,2,2,2,2,3}, with right half (exclusive) {2,2,3}, so a(810) = 7.
The prime indices of 3675 are {2,3,3,4,4}, with right half (exclusive) {4,4}, so a(3675) = 8.
		

Crossrefs

Positions of 0's are 1 and A000040.

Positions of last appearances are A004171.

Positions of first appearances are A100484.

These partitions are counted by A360672.

The value k > 0 appears A360673(k) times, inclusive A360671.

The left version is A360676.

The inclusive version is A360679.

A112798 lists prime indices, length A001222, sum A056239, median* A360005.

A360616 gives half of bigomega (exclusive), inclusive A360617.

First for prime indices, second for partitions, third for prime factors:

- A360676 gives left sum (exclusive), counted by A360672, product A361200.

- A360677 gives right sum (exclusive), counted by A360675, product A361201.

- A360678 gives left sum (inclusive), counted by A360675, product A347043.

- A360679 gives right sum (inclusive), counted by A360672, product A347044.

Cf. A001248, A026424, A359908, A359912, A360006, A360457.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[Take[prix[n],-Floor[Length[prix[n]]/2]]],{n,100}]

Formula

Last position of k is 2^(2k+1).

A360676(n) + A360679(n) = A001222(n).

A360677(n) + A360678(n) = A001222(n).