proposed

approved

editing

proposed

For mean instead of two times median we have A326619/A326620.

This sequence is A360457.

`A326567/A326568 gives mean of prime indices.

A326619/A326620 gives mean of distinct prime indices.

Cf. A000975, A026424, A316413, `A359889, `A359890, A359907, A359908, A359912, A360248, A360453, `A360454.

Positions of first appearances (if we prepend 0 or 1) are A360006, sorted A360007.

A316413 A304038 lists numbers whose distinct prime indices have integer mean.

A325347 = counts partitions w/ with integer median, strict A359907, complement A307683, ranks A359908.

`A326567/A326568 gives mean of prime indices.

Cf. A000975 subs_int_medn, A026424 odd_om, A078174 prifacs_dstnct_mean_int, A304038 dstnct_prix, A359889 prix_mean_eq_medn, A359890 prix_mean_neq_medn, odd A359912 prix_nonint_medn, A360009 prix_int_mean_int_medn, A360248 prix_dstnctprix_diff_medn, A360453 prisig_dstnctprix_same_medn, A360454 prisig_prix_same_medn.

Cf. A000975, A026424, A316413, `A359889, `A359890, A359907, A359908, A359912, A360248, A360453, `A360454.

allocated for Gus WisemanTwo times the median of the set of distinct prime indices of n; a(1) = 1.

1, 2, 4, 2, 6, 3, 8, 2, 4, 4, 10, 3, 12, 5, 5, 2, 14, 3, 16, 4, 6, 6, 18, 3, 6, 7, 4, 5, 20, 4, 22, 2, 7, 8, 7, 3, 24, 9, 8, 4, 26, 4, 28, 6, 5, 10, 30, 3, 8, 4, 9, 7, 32, 3, 8, 5, 10, 11, 34, 4, 36, 12, 6, 2, 9, 4, 38, 8, 11, 6, 40, 3, 42, 13, 5, 9, 9, 4, 44, 4

1,2

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length). Since the denominator is always 1 or 2, the median can be represented as an integer by multiplying by 2.

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. Distinct prime indices are listed by A304038.

The prime indices of 65 are {3,6}, with distinct parts {3,6}, with median 9/2, so a(65) = 9.

The prime indices of 900 are {1,1,2,2,3,3}, with distinct parts {1,2,3}, with median 2, so a(900) = 4.

Table[If[n==1, 1, 2*Median[PrimePi/@First/@FactorInteger[n]]], {n, 100}]

The version for divisors is A063655.

The version for all prime indices is A360005.

Positions of first appearances (if we prepend 0 or 1) are A360006, sorted A360007.

This sequence is A360457.

The version for distinct prime factors is A360458.

The version for all prime factors is A360459.

The version for prime multiplicities is A360460.

Positions of even terms are A360550.

Positions of odd terms are A360551.

The version for 0-prepended differences is A360555.

A112798 lists prime indices, length A001222, sum A056239.

A316413 lists numbers whose prime indices have integer mean.

A325347 = partitions w/ integer median, strict A359907, complement A307683, ranks A359908.

A326567/A326568 gives mean of prime indices.

A326619/A326620 gives mean of distinct prime indices.

A359893 and A359901 count partitions by median, odd-length A359902.

Cf. A000975 subs_int_medn, A026424 odd_om, A078174 prifacs_dstnct_mean_int, A304038 dstnct_prix, A359889 prix_mean_eq_medn, A359890 prix_mean_neq_medn, odd A359912 prix_nonint_medn, A360009 prix_int_mean_int_medn, A360248 prix_dstnctprix_diff_medn, A360453 prisig_dstnctprix_same_medn, A360454 prisig_prix_same_medn.

allocated

nonn

Gus Wiseman, Feb 14 2023

approved

editing

allocated for Gus Wiseman

allocated

approved