cp's OEIS Frontend

A360676 Sum of the left half (exclusive) of the prime indices of n.

%I A360676 #14 Feb 03 2025 02:23:35
%S A360676 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 A360676 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 A360676 2,1,0,2,0,1,2,1,4,1,0,2,4,1,0,2,3,1,2
%N A360676 Sum of the left half (exclusive) of the prime indices of n.
%C A360676 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.
%H A360676 Robert Israel, <a href="/A360676/b360676.txt">Table of n, a(n) for n = 1..10000</a>
%F A360676 A360676(n) + A360679(n) = A001222(n).
%F A360676 A360677(n) + A360678(n) = A001222(n).
%e A360676 The prime indices of 810 are {1,2,2,2,2,3}, with left half (exclusive) {1,2,2}, so a(810) = 5.
%e A360676 The prime indices of 3675 are {2,3,3,4,4}, with left half (exclusive) {2,3}, so a(3675) = 5.
%p A360676 f:= proc(n) local F,i,t;
%p A360676   F:= [seq(numtheory:-pi(t[1])$t[2], t = sort(ifactors(n)[2],(a,b) -> a[1] < b[1]))];
%p A360676   add(F[i],i=1..floor(nops(F)/2))
%p A360676 end proc:
%p A360676 map(f, [$1..100]); # _Robert Israel_, Feb 02 2025
%t A360676 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A360676 Table[Total[Take[prix[n],Floor[Length[prix[n]]/2]]],{n,100}]
%Y A360676 Positions of 0's are 1 and A000040.
%Y A360676 Positions of first appearances are 1 and A001248.
%Y A360676 These partitions are counted by A360675, right A360672.
%Y A360676 A112798 lists prime indices, length A001222, sum A056239, median* A360005.
%Y A360676 A360616 gives half of bigomega (exclusive), inclusive A360617.
%Y A360676 A360673 counts multisets by right sum (exclusive), inclusive A360671.
%Y A360676 First for prime indices, second for partitions, third for prime factors:
%Y A360676 - A360676 gives left sum (exclusive), counted by A360672, product A361200.
%Y A360676 - A360677 gives right sum (exclusive), counted by A360675, product A361201.
%Y A360676 - A360678 gives left sum (inclusive), counted by A360675, product A347043.
%Y A360676 - A360679 gives right sum (inclusive), counted by A360672, product A347044.
%Y A360676 Cf. A026424, A280076, A359912, A360006, A360457, A360674.
%K A360676 nonn
%O A360676 1,9
%A A360676 _Gus Wiseman_, Mar 04 2023