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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.

Regular median Median of prime indices is A360005(n)/2.

For low A067538 counts partitions with integer mean instead of median we have A363943, triangle A363945, ranked by A316413.

For high mean instead of median we have A363944, triangle A363946.

A067538 counts partitions with integer mean, strict A102627, ranked by A316413 (complement A348551).

A362611 counts modes in A363943 gives low mean of prime indices, triangle A362614A363945.

A363944 gives high mean of prime indices, triangle A363946.

Cf. A025065, `A026905, A215366, A326567/A326568, A344296, A359889, A359908, `A363723, `A363724, A363727, `A363729, A363740, `A363949, `A363950.

mell[y_]:=If[Length[y]==0, 0, If[OddQ[Length[y]], y[[(Length[y]+1)/2]], y[[Length[y]/2]]]];

Positions of 1's are A363488.

The low median (see A124943) in a multiset is either the middle part (for odd length), or the least of the two middle parts (for even length).

allocated for Gus WisemanLow median in the multiset of prime indices of n.

0, 1, 2, 1, 3, 1, 4, 1, 2, 1, 5, 1, 6, 1, 2, 1, 7, 2, 8, 1, 2, 1, 9, 1, 3, 1, 2, 1, 10, 2, 11, 1, 2, 1, 3, 1, 12, 1, 2, 1, 13, 2, 14, 1, 2, 1, 15, 1, 4, 3, 2, 1, 16, 2, 3, 1, 2, 1, 17, 1, 18, 1, 2, 1, 3, 2, 19, 1, 2, 3, 20, 1, 21, 1, 3, 1, 4, 2, 22, 1, 2, 1

1,3

The low median in a multiset is either the middle part (for odd length), or the least of the two middle parts (for even length).

The prime indices of 90 are {1,2,2,3}, with low median 2, so a(90) = 2.

The prime indices of 150 are {1,2,3,3}, with low median 2, so a(150) = 2.

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

mell[y_]:=If[Length[y]==0, 0, If[OddQ[Length[y]], y[[(Length[y]+1)/2]], y[[Length[y]/2]]]];

Table[mell[prix[n]], {n, 30}]

Positions of first appearances are 1 and A000040.

The triangle for this statistic (low median) is A124943, high A124944.

Regular median of prime indices is A360005(n)/2.

For mode instead of median we have A363486, high A363487.

The high version is A363942.

For low mean instead of median we have A363943, triangle A363945.

For high mean instead of median we have A363944, triangle A363946.

A067538 counts partitions with integer mean, strict A102627, ranked by A316413 (complement A348551).

A112798 lists prime indices, length A001222, sum A056239.

A362611 counts modes in prime indices, triangle A362614.

Cf. A025065, `A026905, A215366, A326567/A326568, A344296, A359889, A359908, `A363723, `A363724, A363727, `A363729, A363740, `A363949, `A363950.

allocated

nonn

Gus Wiseman, Jul 01 2023

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allocated for Gus Wiseman

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