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%I #6 Mar 07 2023 22:10:21
%S 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,
%T 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,
%U 3,3,19,2,2,4,20,3,21,1,5,2,4,3,22,3,4,1
%N Sum of the left half (inclusive) 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 (inclusive) {1,2,2}, so a(810) = 5.
%e The prime indices of 3675 are {2,3,3,4,4}, with left half (inclusive) {2,3,3}, so a(3675) = 8.
%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Table[Total[Take[prix[n],Ceiling[Length[prix[n]]/2]]],{n,100}]
%Y Positions of first appearances are 1 and A001248.
%Y Positions of 1's are A001747.
%Y These partitions are counted by A360675 with rows reversed.
%Y The exclusive version is A360676.
%Y The right version is A360679.
%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.
%K nonn
%O 1,3
%A _Gus Wiseman_, Mar 05 2023