proposed

approved

editing

proposed

Cf. A046682, `A061395, `A071178, `A316413, A321648, A325040, A326567/A326568, A326848, `A352486, A362617, A362618, `A362980, A363223, `A363224.

Cf. A000720 pi, A046682, A061395 max_prix, A071178 prisig_last, A316413 h_int_avg, A321648 perms_conj_h, A325040 h_sameprod_as_conj, A326567/A326568 h_avg, A326836 h_max_divs_sum, A326845 h_sum_times_max, A326848 h_sqr_mult_of_n, A352486 h_not_selfconj, A362617 prix_not_unq_medn, A362618 prix_unq_medn, A362621 prix_medn_eq_max, A362622 prix_max_in_mid, A362980 prix_medn_neq_max, A362983 num_prifacs_grtr_min, A363133 h_twi_min_eq_mean, A363134 om_eq_twi_minprix, A363218 om_eq_twi_maxprix, A363222 om_eq_max_minus_min, A363223 om_eq_medn, A363224 mean_eq_twi_medn, A363260 ptns_pts_dsjnt_diff.

Cf. A046682, `A061395, `A071178, `A316413, A321648, A325040, A326567/A326568, A326848, `A352486, A362617, A362618, `A362980, A363223, `A363224.

This sequence is A363219the conjugate version of A360005.

A003963 = product of prime indices, conjugate A329382.

A122111 = is partition conjugation using in terms of Heinz numbers.

A352487 gives excedance set (counted by A000701), weak A352489 (counted by A046682).

A352490 gives nonexcedance set (counted by A000701), weak A352488 (counted by A046682).

Cf. A000720 pi, A046682, A061395 max_prix, A071178 prisig_last, A316413 h_int_avg, A321648 perms_conj_h, A325040 h_sameprod_as_conj, A326567/A326568 h_avg, A326836 h_max_divs_sum, A326845 h_sum_times_max, A326848 h_sqr_mult_of_n, A352486 h_not_selfconj, A362617 prix_not_unq_medn, A362618 prix_unq_medn, A362621 prix_medn_eq_max, A362622 prix_max_in_mid, A362980 prix_medn_neq_max, A362983 num_prifacs_grtr_min, A363133 h_twi_min_eq_mean, A363134 om_eq_twi_minprix, A363218 om_eq_twi_maxprix, A363222 om_eq_max_minus_min, A363223 om_eq_medn, A363224 mean_eq_twi_medn, A363260 ptns_pts_dsjnt_diff.

allocated for Gus WisemanTwice the median of the conjugate of the integer partition with Heinz number n.

0, 2, 2, 4, 2, 3, 2, 6, 4, 2, 2, 4, 2, 2, 4, 8, 2, 5, 2, 2, 3, 2, 2, 5, 4, 2, 6, 2, 2, 4, 2, 10, 2, 2, 4, 6, 2, 2, 2, 2, 2, 3, 2, 2, 6, 2, 2, 6, 4, 4, 2, 2, 2, 7, 4, 2, 2, 2, 2, 4, 2, 2, 4, 12, 3, 2, 2, 2, 2, 4, 2, 7, 2, 2, 6, 2, 4, 2, 2, 2, 8, 2, 2, 3, 2, 2

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.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

The partition (4,2,1) has Heinz number 42 and conjugate (3,2,1,1) with median 3/2, so a(42) = 3.

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

conj[y_]:=If[Length[y]==0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]];

Table[If[n==1, 0, 2*Median[conj[prix[n]]]], {n, 100}]

Twice the row media of A321649 or A321650.

For mean instead of twice median we have A326839/A326840.

This sequence is A363219.

A000700 counts self-conjugate partitions, ranked by A088902 (cf. A258116).

A003963 = product of prime indices, conjugate A329382.

A056239 adds up prime indices, row sums of A112798 and A296150.

A122111 = partition conjugation using Heinz numbers.

A124010 gives prime signature, sorted A118914, length A001221, sum A001222.

A352487 gives excedance set (counted by A000701), weak A352489 (counted by A046682).

A352490 gives nonexcedance set (counted by A000701), weak A352488 (counted by A046682).

A352491 gives n minus Heinz number of conjugate.

A363220 counts partitions with same median as conjugate.

Cf. A000720 pi, A061395 max_prix, A071178 prisig_last, A316413 h_int_avg, A321648 perms_conj_h, A325040 h_sameprod_as_conj, A326567/A326568 h_avg, A326836 h_max_divs_sum, A326845 h_sum_times_max, A326848 h_sqr_mult_of_n, A352486 h_not_selfconj, A362617 prix_not_unq_medn, A362618 prix_unq_medn, A362621 prix_medn_eq_max, A362622 prix_max_in_mid, A362980 prix_medn_neq_max, A362983 num_prifacs_grtr_min, A363133 h_twi_min_eq_mean, A363134 om_eq_twi_minprix, A363218 om_eq_twi_maxprix, A363222 om_eq_max_minus_min, A363223 om_eq_medn, A363224 mean_eq_twi_medn, A363260 ptns_pts_dsjnt_diff.

allocated

nonn

Gus Wiseman, May 25 2023

approved

editing

allocated for Gus Wiseman

allocated

approved