%I #9 Mar 07 2023 22:10:27

%S 0,0,0,1,0,1,0,1,2,1,0,1,0,1,2,2,0,1,0,1,2,1,0,2,3,1,2,1,0,1,0,2,2,1,

%T 3,2,0,1,2,2,0,1,0,1,2,1,0,2,4,1,2,1,0,3,3,2,2,1,0,2,0,1,2,3,3,1,0,1,

%U 2,1,0,2,0,1,2,1,4,1,0,2,4,1,0,2,3,1,2

%N Sum of the left half (exclusive) of the prime indices of n.

%C 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.

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

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

%e The prime indices of 810 are {1,2,2,2,2,3}, with left half (exclusive) {1,2,2}, so a(810) = 5.

%e The prime indices of 3675 are {2,3,3,4,4}, with left half (exclusive) {2,3}, so a(3675) = 5.

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Table[Total[Take[prix[n],Floor[Length[prix[n]]/2]]],{n,100}]

%Y Positions of 0's are 1 and A000040.

%Y Positions of first appearances are 1 and A001248.

%Y These partitions are counted by A360675, right A360672.

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

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

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

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

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

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

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

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

%Y Cf. A026424, A280076, A359912, A360006, A360457, A360674.

%K nonn

%O 1,9

%A _Gus Wiseman_, Mar 04 2023