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The partition (3,2,1,1) has 4 parts {1,2,3,4} and 3 distinct divisors of parts {1,2,3}, so is counted under a(7).

These partitions have The Heinz numbers of these partitions are given by A370348.

For submultisets instead of parts on the LHS we get new, ranks A371167.

Other equalities: A371172 (A371165), A371178 (A371177).

A355731 counts choices of a divisor of each prime index, firsts A355732.

`A370814 counts divisor-choosable factorizations, complement A370813.

Cf. A003963, `A066739, A319055, ~A322527, ~A322530, `A355737, `A355739, `A355741, A355731, A370803, A370808, A370809, `A371166, `A371169, `A371170A370813, A370814.

A239312 A355731 counts choices of a divisor-choosable partitions, ranks A368110 of each prime index, firsts A355732.

`~A355731 counts choices of a divisor of each prime index, firsts A355732.

Choosable partitions: A239312 (A368110), A355740 (A370320), A370592 (A368100), A370593 (A355529).

A355740 `A370814 counts non-divisor-choosable partitions, ranks A370320factorizations, complement A370813.

A370592 counts factor-choosable partitions, ranks A368100.

A370593 counts non-factor-choosable partitions, ranks A355529.

A370813 counts non-divisor-choosable factorizations, complement A370814.

Cf. A000792 nex_ones_max, A003963 h_prod, A014499 bpe_pri, A048249 nex_ones, A064573 ptns_use_pow_samepri, A066739 sum_of_prod, A319055 maxprod_ptn_relpri, A319616 co_bal_bmp, A319877 prix_prod_sqr_sqf, A320325 prix_prod_perpow, A322527 ptns_prod_pow_sqf, A322530 ptns_no1_prod_sqf, A355737 choose_div_each_prix_so_relpri, A355739 choose_div_each_prix_so_strict, A355741 choose_prifac_each_prix, A370803 ptns_mult_ways_diff_div_each, A370808 max_num_choices_div_each_pt_ptn, A370809 max_num_choices_prifac_each_pt_ptn, A371166 divs_less_prixdivs, A371169 leq_div_prix_prix, A371170 geq_div_prix_prix.

Cf. A003963, `A066739, A319055, ~A322527, ~A322530, `A355737, `A355739, `A355741, A370803, A370808, A370809, `A371166, `A371169, `A371170.

allocated for Gus WisemanNumber of integer partitions of n with more parts than distinct divisors of parts.

0, 0, 1, 1, 2, 4, 5, 9, 12, 18, 26, 34, 50, 65, 92, 121, 161, 209, 274, 353, 456, 590, 745, 950, 1195, 1507, 1885, 2350, 2923, 3611, 4465, 5485, 6735, 8223, 10050, 12195, 14822, 17909, 21653, 26047, 31340, 37557, 44990, 53708, 64068, 76241, 90583, 107418

1,5

These partitions have Heinz numbers A370348.

The a(0) = 0 through a(8) = 12 partitions:

. . (11) (111) (211) (221) (222) (331) (2222)

(1111) (311) (2211) (511) (3221)

(2111) (3111) (2221) (3311)

(11111) (21111) (3211) (4211)

(111111) (4111) (5111)

(22111) (22211)

(31111) (32111)

(211111) (41111)

(1111111) (221111)

(311111)

(2111111)

(11111111)

Table[Length[Select[IntegerPartitions[n], Length[#] > Length[Union@@Divisors/@#]&]], {n, 0, 30}]

The partitions are ranked by A370348.

The opposite version is A371173, ranked by A371168.

The RHS is represented by A370820, positions of twos A371127.

The version for equality is A371130 (ranks A370802), strict A371128.

For submultisets instead of parts on the LHS we get new, ranks A371167.

Other equalities: A371172 (A371165), A371178 (A371177).

A000005 counts divisors.

A239312 counts divisor-choosable partitions, ranks A368110.

`~A355731 counts choices of a divisor of each prime index, firsts A355732.

A355740 counts non-divisor-choosable partitions, ranks A370320.

A370592 counts factor-choosable partitions, ranks A368100.

A370593 counts non-factor-choosable partitions, ranks A355529.

A370813 counts non-divisor-choosable factorizations, complement A370814.

Cf. A000792 nex_ones_max, A003963 h_prod, A014499 bpe_pri, A048249 nex_ones, A064573 ptns_use_pow_samepri, A066739 sum_of_prod, A319055 maxprod_ptn_relpri, A319616 co_bal_bmp, A319877 prix_prod_sqr_sqf, A320325 prix_prod_perpow, A322527 ptns_prod_pow_sqf, A322530 ptns_no1_prod_sqf, A355737 choose_div_each_prix_so_relpri, A355739 choose_div_each_prix_so_strict, A355741 choose_prifac_each_prix, A370803 ptns_mult_ways_diff_div_each, A370808 max_num_choices_div_each_pt_ptn, A370809 max_num_choices_prifac_each_pt_ptn, A371166 divs_less_prixdivs, A371169 leq_div_prix_prix, A371170 geq_div_prix_prix.

allocated

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Gus Wiseman, Mar 16 2024

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editing

allocated for Gus Wiseman

allocated

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