A360678 Sum of the left half (inclusive) of the prime indices of n.

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

Offset

1,3

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.

Links

Table of n, a(n) for n=1..82.

Formula

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

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

Example

The prime indices of 810 are {1,2,2,2,2,3}, with left half (inclusive) {1,2,2}, so a(810) = 5.
The prime indices of 3675 are {2,3,3,4,4}, with left half (inclusive) {2,3,3}, so a(3675) = 8.

Mathematica

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

Crossrefs

Positions of first appearances are 1 and A001248.

Positions of 1's are A001747.

These partitions are counted by A360675 with rows reversed.

The exclusive version is A360676.

The right version is A360679.

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

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

A360673 counts multisets by right sum (exclusive), inclusive A360671.

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. A026424, A280076, A359912, A360006, A360457.

Sequence in context: A067734 A303758 A161904 * A346697 A338565 A323638

Adjacent sequences: A360675 A360676 A360677 * A360679 A360680 A360681

Keyword

nonn

Author

Gus Wiseman, Mar 05 2023

Status

approved