A370803
Number of integer partitions of n such that more than one set can be obtained by choosing a different divisor of each part.
Original entry on oeis.org
0, 0, 1, 1, 1, 3, 2, 4, 5, 7, 10, 11, 15, 18, 25, 28, 39, 45, 59, 66, 83, 101, 123, 150, 176, 213, 252, 301, 352, 426, 497, 589, 684, 802, 939, 1095, 1270, 1480, 1718, 1985, 2289, 2645, 3056, 3489, 4019, 4590, 5289, 6014, 6877, 7817, 8955, 10134, 11551, 13085
Offset: 0
The partition (6,4,4,1) has two choices, namely {1,2,4,6} and {1,2,3,4}, so is counted under a(15).
The a(0) = 0 through a(13) = 18 partitions (A..D = 10..13):
. . 2 3 4 5 6 7 8 9 A B C D
32 42 43 44 54 64 65 66 76
41 52 53 63 73 74 75 85
61 62 72 82 83 84 94
431 81 91 92 93 A3
432 433 A1 A2 B2
621 532 443 543 C1
541 542 633 544
622 632 642 643
631 641 651 652
821 732 661
741 742
822 832
831 841
921 922
A21
5431
6421
Including partitions with one choice gives
A239312, complement
A370320.
These partitions have ranks
A370811.
A355731 counts choices of a divisor of each prime index, firsts
A355732.
A355733 counts divisor-choices of prime indices.
-
Table[Length[Select[IntegerPartitions[n],Length[Union[Sort /@ Select[Tuples[Divisors/@#],UnsameQ@@#&]]]>1&]],{n,0,30}]
A371130
Number of integer partitions of n such that the number of parts is equal to the number of distinct divisors of parts.
Original entry on oeis.org
1, 1, 0, 1, 2, 0, 4, 2, 4, 5, 5, 11, 10, 16, 17, 21, 26, 32, 44, 53, 69, 71, 101, 110, 148, 168, 205, 249, 289, 356, 418, 502, 589, 716, 812, 999, 1137, 1365, 1566, 1873, 2158, 2537, 2942, 3449, 4001, 4613, 5380, 6193, 7220, 8224, 9575, 10926, 12683, 14430
Offset: 0
The partition (6,2,2,1) has 4 parts and 4 distinct divisors of parts {1,2,3,6} so is counted under a(11).
The a(1) = 1 through a(11) = 11 partitions:
(1) . (21) (22) . (33) (322) (71) (441) (55) (533)
(31) (51) (421) (332) (522) (442) (722)
(321) (422) (531) (721) (731)
(411) (521) (4311) (4321) (911)
(6111) (6211) (4322)
(4331)
(5321)
(5411)
(6221)
(6311)
(8111)
These partitions are ranked by
A370802.
For (greater than) instead of (equal to) we have
A371171, ranks
A370348.
For submultisets instead of parts on the LHS we have
A371172.
For (less than) instead of (equal to) we have
A371173, ranked by
A371168.
A008284 counts partitions by length.
-
Table[Length[Select[IntegerPartitions[n], Length[#]==Length[Union@@Divisors/@#]&]],{n,0,30}]
A370586
Number of subsets of {1..n} containing n such that it is possible to choose a different prime factor of each element (choosable).
Original entry on oeis.org
0, 0, 1, 2, 2, 6, 8, 20, 12, 20, 44, 116, 88, 320, 380, 508, 264, 1792, 968, 4552, 3136, 5600, 10056, 27896, 11792, 16384, 46688, 19584, 48288, 198528, 110928, 507984, 99648, 463552, 859376, 821136, 470688, 3730368, 4033920, 4651296, 2932512, 19078464
Offset: 0
The a(0) = 0 through a(7) = 20 subsets:
. . {2} {3} {4} {5} {6} {7}
{2,3} {3,4} {2,5} {2,6} {2,7}
{3,5} {3,6} {3,7}
{4,5} {4,6} {4,7}
{2,3,5} {5,6} {5,7}
{3,4,5} {2,5,6} {6,7}
{3,5,6} {2,3,7}
{4,5,6} {2,5,7}
{2,6,7}
{3,4,7}
{3,5,7}
{3,6,7}
{4,5,7}
{4,6,7}
{5,6,7}
{2,3,5,7}
{2,5,6,7}
{3,4,5,7}
{3,5,6,7}
{4,5,6,7}
Maximal choosable sets are counted by
A370585.
The complement is counted by
A370587.
For a unique choice we have
A370588.
For binary indices instead of prime factors we have
A370639.
A355741 counts choices of a prime factor of each prime index.
A368098 counts choosable unlabeled multiset partitions, complement
A368097.
Cf.
A000040,
A000720,
A005117,
A045778,
A133686,
A355739,
A355744,
A355745,
A367771,
A367905,
A370636.
-
Table[Length[Select[Subsets[Range[n]], MemberQ[#,n]&&Length[Select[Tuples[If[#==1, {},First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]>0&]],{n,0,10}]
A370594
Number of integer partitions of n such that only one set can be obtained by choosing a different prime factor of each part.
Original entry on oeis.org
1, 0, 1, 1, 1, 2, 0, 3, 3, 4, 3, 4, 5, 5, 8, 10, 11, 7, 14, 13, 19, 23, 24, 20, 30, 33, 40, 47, 49, 55, 53, 72, 80, 90, 92, 110, 110, 132, 154, 169, 180, 201, 218, 246, 281, 302, 323, 348, 396, 433, 482, 530, 584, 618, 670, 754, 823, 903, 980, 1047, 1137
Offset: 0
The partition (10,6,4) has unique choice (5,3,2) so is counted under a(20).
The a(0) = 1 through a(12) = 5 partitions:
() . (2) (3) (4) (5) . (7) (8) (9) (6,4) (11) (6,6)
(3,2) (4,3) (5,3) (5,4) (7,3) (7,4) (7,5)
(5,2) (6,2) (6,3) (5,3,2) (8,3) (10,2)
(7,2) (9,2) (5,4,3)
(7,3,2)
Maximal sets of this type are counted by
A370585.
For divisors instead of factors we have
A370595.
These partitions have ranks
A370647.
A355741 counts ways to choose a prime factor of each prime index.
-
Table[Length[Select[IntegerPartitions[n], Length[Union[Sort/@Select[Tuples[If[#==1, {},First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]]==1&]],{n,0,30}]
A371171
Number of integer partitions of n with more parts than distinct divisors of parts.
Original entry on oeis.org
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
Offset: 1
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).
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)
The partitions are ranked by
A370348.
For submultisets instead of parts on the LHS we get ranks
A371167.
-
Table[Length[Select[IntegerPartitions[n],Length[#] > Length[Union@@Divisors/@#]&]],{n,0,30}]
A370595
Number of integer partitions of n such that only one set can be obtained by choosing a different divisor of each part.
Original entry on oeis.org
1, 1, 0, 1, 2, 0, 3, 2, 4, 3, 4, 5, 8, 9, 8, 13, 12, 17, 16, 27, 28, 33, 36, 39, 50, 58, 65, 75, 93, 94, 112, 125, 148, 170, 190, 209, 250, 273, 305, 341, 403, 432, 484, 561, 623, 708, 765, 873, 977, 1109, 1178, 1367, 1493, 1669, 1824, 2054, 2265, 2521, 2770
Offset: 0
The a(1) = 1 through a(15) = 13 partitions (A = 10, B = 11, C = 12, D = 13):
1 . 21 22 . 33 322 71 441 55 533 B1 553 77 933
31 51 421 332 522 442 722 444 733 D1 B22
321 422 531 721 731 552 751 B21 B31
521 4321 4322 4332 931 4433 4443
5321 4431 4432 5441 5442
5322 5332 6332 5532
5421 5422 7322 6621
6321 6322 7421 7332
7321 7422
7521
8421
9321
54321
The version for prime factors (not all divisors) is
A370594, ranks
A370647.
These partitions have ranks
A370810.
A355731 counts choices of a divisor of each prime index, firsts
A355732.
A370592 counts partitions with choosable prime factors, ranks
A368100.
A370593 counts partitions without choosable prime factors, ranks
A355529.
A370804 counts non-condensed partitions with no ones, complement
A370805.
A370814 counts factorizations with choosable divisors, complement
A370813.
-
Table[Length[Select[IntegerPartitions[n],Length[Union[Sort /@ Select[Tuples[Divisors/@#],UnsameQ@@#&]]]==1&]],{n,0,30}]
A371168
Positive integers with fewer prime factors (A001222) than distinct divisors of prime indices (A370820).
Original entry on oeis.org
3, 5, 7, 11, 13, 14, 15, 17, 19, 21, 23, 26, 29, 31, 33, 35, 37, 38, 39, 41, 43, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 65, 67, 69, 70, 71, 73, 74, 76, 77, 78, 79, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 105, 106, 107, 109, 111, 113, 114, 115
Offset: 1
The prime indices of 105 are {2,3,4}, and there are 3 prime factors (3,5,7) and 4 distinct divisors of prime indices (1,2,3,4), so 105 is in the sequence.
The terms together with their prime indices begin:
3: {2} 35: {3,4} 59: {17} 86: {1,14}
5: {3} 37: {12} 61: {18} 87: {2,10}
7: {4} 38: {1,8} 65: {3,6} 89: {24}
11: {5} 39: {2,6} 67: {19} 91: {4,6}
13: {6} 41: {13} 69: {2,9} 93: {2,11}
14: {1,4} 43: {14} 70: {1,3,4} 94: {1,15}
15: {2,3} 46: {1,9} 71: {20} 95: {3,8}
17: {7} 47: {15} 73: {21} 97: {25}
19: {8} 49: {4,4} 74: {1,12} 101: {26}
21: {2,4} 51: {2,7} 76: {1,1,8} 103: {27}
23: {9} 52: {1,1,6} 77: {4,5} 105: {2,3,4}
26: {1,6} 53: {16} 78: {1,2,6} 106: {1,16}
29: {10} 55: {3,5} 79: {22} 107: {28}
31: {11} 57: {2,8} 83: {23} 109: {29}
33: {2,5} 58: {1,10} 85: {3,7} 111: {2,12}
For divisors instead of prime factors on the LHS we get
A371166.
The complement is counted by
A371169.
Partitions of this type are counted by
A371173.
A001221 counts distinct prime factors.
A355731 counts choices of a divisor of each prime index, firsts
A355732.
A371173
Number of integer partitions of n with fewer parts than distinct divisors of parts.
Original entry on oeis.org
0, 0, 1, 1, 1, 3, 2, 4, 6, 7, 11, 11, 17, 20, 26, 34, 44, 56, 67, 84, 102, 131, 156, 195, 232, 283, 346, 411, 506, 598, 721, 855, 1025, 1204, 1448, 1689, 2018, 2363, 2796, 3265, 3840, 4489, 5242, 6104, 7106, 8280, 9595, 11143, 12862, 14926, 17197, 19862, 22841
Offset: 0
The partition (4,3,2) has 3 parts {2,3,4} and 4 distinct divisors of parts {1,2,3,4}, so is counted under a(9).
The a(2) = 1 through a(10) = 11 partitions:
(2) (3) (4) (5) (6) (7) (8) (9) (10)
(3,2) (4,2) (4,3) (4,4) (5,4) (6,4)
(4,1) (5,2) (5,3) (6,3) (7,3)
(6,1) (6,2) (7,2) (8,2)
(4,3,1) (8,1) (9,1)
(6,1,1) (4,3,2) (4,3,3)
(6,2,1) (5,3,2)
(5,4,1)
(6,2,2)
(6,3,1)
(8,1,1)
For submultisets instead of parts on the LHS we get ranks
A371166.
These partitions are ranked by
A371168.
A355731 counts choices of a divisor of each prime index, firsts
A355732.
-
Table[Length[Select[IntegerPartitions[n],Length[#] < Length[Union@@Divisors/@#]&]],{n,0,30}]
A370587
Number of subsets of {1..n} containing n such that it is not possible to choose a different prime factor of each element (non-choosable).
Original entry on oeis.org
0, 1, 1, 2, 6, 10, 24, 44, 116, 236, 468, 908, 1960, 3776, 7812, 15876, 32504, 63744, 130104, 257592, 521152, 1042976, 2087096, 4166408, 8376816, 16760832, 33507744, 67089280, 134169440, 268236928, 536759984, 1073233840, 2147384000, 4294503744, 8589075216, 17179048048
Offset: 0
The a(0) = 0 through a(5) = 10 subsets:
. {1} {1,2} {1,3} {1,4} {1,5}
{1,2,3} {2,4} {1,2,5}
{1,2,4} {1,3,5}
{1,3,4} {1,4,5}
{2,3,4} {2,4,5}
{1,2,3,4} {1,2,3,5}
{1,2,4,5}
{1,3,4,5}
{2,3,4,5}
{1,2,3,4,5}
The complement is counted by
A370586.
For a unique choice we have
A370588.
For binary indices instead of factors we have
A370639, complement
A370589.
A355741 counts choices of a prime factor of each prime index.
A368098 counts choosable unlabeled multiset partitions, complement
A368097.
A370585 counts maximal choosable sets.
-
Table[Length[Select[Subsets[Range[n]],MemberQ[#,n] && Length[Select[Tuples[If[#==1,{},First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]==0&]],{n,0,10}]
A370810
Numbers n such that only one set can be obtained by choosing a different divisor of each prime index of n.
Original entry on oeis.org
1, 2, 6, 9, 10, 22, 25, 30, 34, 42, 45, 62, 63, 66, 75, 82, 98, 99, 102, 110, 118, 121, 134, 147, 153, 166, 170, 186, 210, 218, 230, 246, 254, 275, 279, 289, 310, 314, 315, 330, 343, 354, 358, 363, 369, 374, 382, 390, 402, 410, 422, 425, 462, 482, 490, 495
Offset: 1
The prime indices of 6591 are {2,6,6,6}, for which the only choice is {1,2,3,6}, so 6591 is in the sequence.
The terms together with their prime indices begin:
1: {}
2: {1}
6: {1,2}
9: {2,2}
10: {1,3}
22: {1,5}
25: {3,3}
30: {1,2,3}
34: {1,7}
42: {1,2,4}
45: {2,2,3}
62: {1,11}
63: {2,2,4}
66: {1,2,5}
75: {2,3,3}
82: {1,13}
98: {1,4,4}
99: {2,2,5}
102: {1,2,7}
110: {1,3,5}
A355731 counts choices of a divisor of each prime index, firsts
A355732.
A370814 counts factorizations with choosable divisors, complement
A370813.
-
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Length[Union[Sort /@ Select[Tuples[Divisors/@prix[#]],UnsameQ@@#&]]]==1&]
Comments