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A363489 Rounded mean of the multiset of prime indices of n.

%I A363489 #10 Jul 08 2023 23:06:29
%S A363489 0,1,2,1,3,2,4,1,2,2,5,1,6,2,2,1,7,2,8,2,3,3,9,1,3,4,2,2,10,2,11,1,4,
%T A363489 4,4,2,12,4,4,2,13,2,14,2,2,5,15,1,4,2,4,3,16,2,4,2,5,6,17,2,18,6,3,1,
%U A363489 4,3,19,3,6,3,20,1,21,6,3,3,4,3,22,1,2,7
%N A363489 Rounded mean of the multiset of prime indices of n.
%C A363489 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.
%C A363489 We use the "rounding half to even" rule, see link.
%H A363489 Wikipedia, <a href="https://en.wikipedia.org/wiki/Rounding">Rounding</a>.
%e A363489 The prime indices of 180 are {1,1,2,2,3}, with mean 9/5, which rounds to 2, so a(180) = 2.
%t A363489 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A363489 Table[If[n==1,0,Round[Mean[prix[n]]]],{n,100}]
%Y A363489 Positions of first appearances are 1 and A000040.
%Y A363489 Before rounding we had A326567/A326568.
%Y A363489 For rounded-down: A363943, triangle A363945.
%Y A363489 For rounded-up: A363944, triangle A363946.
%Y A363489 Positions of 1's are A363948, complement A364059.
%Y A363489 The triangle for this statistic (rounded mean) is A364060.
%Y A363489 For prime factors instead of indices we have A364061.
%Y A363489 A088529/A088530 gives mean of prime signature A124010.
%Y A363489 A112798 lists prime indices, length A001222, sum A056239.
%Y A363489 A316413 ranks partitions with integer mean, counted by A067538.
%Y A363489 Cf. A327473, A327476, A327482, A359889, A360005, A363947, A363949, A363951.
%K A363489 nonn
%O A363489 1,3
%A A363489 _Gus Wiseman_, Jul 07 2023