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Sum of the right half (inclusive) of the prime indices of n.
17

%I #6 Mar 07 2023 22:10:16

%S 0,1,2,1,3,2,4,2,2,3,5,3,6,4,3,2,7,4,8,4,4,5,9,3,3,6,4,5,10,5,11,3,5,

%T 7,4,4,12,8,6,4,13,6,14,6,5,9,15,4,4,6,7,7,16,4,5,5,8,10,17,5,18,11,6,

%U 3,6,7,19,8,9,7,20,5,21,12,6,9,5,8,22,5,4

%N Sum of the right 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 right half (inclusive) {2,2,3}, so a(810) = 7.

%e The prime indices of 3675 are {2,3,3,4,4}, with right half (inclusive) {3,4,4}, so a(3675) = 11.

%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 The value k appears A360671(k) times, exclusive A360673.

%Y These partitions are counted by A360672 with rows reversed.

%Y The exclusive version is A360677.

%Y The left version is A360678.

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

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

%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, A359912, A360006, A360007, A360457.

%K nonn

%O 1,3

%A _Gus Wiseman_, Mar 05 2023