A363944 - OEIS

%I #9 Jul 01 2023 20:54:12

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

%T 4,4,2,12,5,4,2,13,3,14,3,3,5,15,2,4,3,5,3,16,2,4,2,5,6,17,2,18,6,3,1,

%U 5,3,19,3,6,3,20,2,21,7,3,4,5,3,22,2,2,7

%N Mean of the multiset of prime indices of n, rounded up.

%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.

%C Extending the terminology introduced at A124944, this is the "high mean" of prime indices.

%e The prime indices of 360 are {1,1,1,2,2,3}, with mean 3/2, so a(360) = 2.

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

%t meanup[y_]:=If[Length[y]==0,0,Ceiling[Mean[y]]];

%t Table[meanup[prix[n]],{n,100}]

%Y Positions of first appearances are 1 and A000040.

%Y Positions of 1's are A000079(n>0).

%Y Before rounding up we had A326567/A326568.

%Y For mode instead of mean we have A363487, low A363486.

%Y For median instead of mean we have A363942, triangle A124944.

%Y Rounding down instead of up gives A363943, triangle A363945.

%Y The triangle for this statistic (high mean) is A363946.

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

%Y A316413 ranks partitions with integer mean, counted by A067538.

%Y A360005 gives twice the median of prime indices.

%Y A363947 ranks partitions with rounded mean 1, counted by A363948.

%Y A363949 ranks partitions with low mean 1, counted by A025065.

%Y A363950 ranks partitions with low mean 2, counted by A026905 redoubled.

%Y Cf. A051293, A124943, A215366, A327473, A327476, A327482, A359889, A362611, A363723, A363724, A363727, A363951.

%K nonn

%O 1,3

%A _Gus Wiseman_, Jun 30 2023