A319179
Number of integer partitions of n that are relatively prime but not aperiodic. Number of integer partitions of n that are aperiodic but not relatively prime.
Original entry on oeis.org
0, 1, 1, 1, 1, 2, 1, 3, 2, 6, 1, 9, 1, 14, 7, 17, 1, 32, 1, 36, 15, 55, 1, 77, 6, 100, 27, 121, 1, 200, 1, 209, 56, 296, 19, 403, 1, 489, 101, 596, 1, 885, 1, 947, 192, 1254, 1, 1673, 14, 1979, 297, 2336, 1, 3300, 60, 3594, 490, 4564, 1, 5988, 1, 6841, 800
Offset: 1
The a(12) = 9 integer partitions that are relatively prime but not aperiodic:
(5511),
(332211), (333111), (441111),
(22221111), (33111111),
(222111111),
(2211111111),
(111111111111).
The a(12) = 9 integer partitions that are aperiodic but not relatively prime:
(12),
(8,4), (9,3), (10,2),
(6,3,3), (6,4,2), (8,2,2),
(6,2,2,2),
(4,2,2,2,2).
-
Table[Length[Select[IntegerPartitions[n],And[GCD@@#==1,GCD@@Length/@Split[#]>1]&]],{n,30}]
A325332
Number of totally abnormal integer partitions of n.
Original entry on oeis.org
0, 0, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 8, 1, 7, 5, 10, 2, 16, 4, 21, 15, 24, 17, 49, 29, 53, 53, 84, 65, 121, 92, 148, 141, 186, 179, 280, 223, 317, 318, 428, 387, 576, 512, 700, 734, 899, 900, 1260, 1207, 1551, 1668, 2041, 2109, 2748, 2795, 3463, 3775, 4446
Offset: 0
The a(2) = 1 through a(12) = 8 totally abnormal partitions (A = 10, B = 11, C = 12):
(2) (3) (4) (5) (6) (7) (8) (9) (A) (B) (C)
(22) (33) (44) (333) (55) (66)
(222) (2222) (3322) (444)
(3311) (4411) (3333)
(22222) (4422)
(5511)
(222222)
(333111)
-
normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
antinrmQ[ptn_]:=!normQ[ptn]&&(Length[ptn]==1||antinrmQ[Sort[Length/@Split[ptn]]]);
Table[Length[Select[IntegerPartitions[n],antinrmQ]],{n,0,30}]
A319811
Number of totally aperiodic integer partitions of n.
Original entry on oeis.org
1, 1, 2, 3, 6, 7, 14, 17, 27, 34, 55, 63, 99, 117, 162, 203, 286, 333, 469, 558, 737, 903, 1196, 1414, 1860, 2232, 2839, 3422, 4359, 5144, 6531, 7762, 9617, 11479, 14182, 16715, 20630, 24333, 29569, 34890, 42335, 49515, 59871, 70042, 83810, 98105, 117152
Offset: 1
The a(6) = 7 aperiodic integer partitions are: (6), (51), (42), (411), (321), (3111), (21111). The first aperiodic integer partition that is not totally aperiodic is (432211).
Cf.
A000837,
A018783,
A047966,
A098859,
A100953,
A305563,
A319149,
A319160,
A319162,
A319163,
A319164,
A319810.
-
totaperQ[m_]:=Or[UnsameQ@@m,And[GCD@@Length/@Split[Sort[m]]==1,totaperQ[Sort[Length/@Split[Sort[m]]]]]];
Table[Length[Select[IntegerPartitions[n],totaperQ]],{n,30}]
A319153
Number of integer partitions of n that reduce to 2, meaning their Heinz number maps to 2 under A304464.
Original entry on oeis.org
0, 2, 1, 3, 5, 7, 12, 17, 24, 33, 44, 57, 76, 100, 129, 168, 214, 282, 355, 462, 586, 755, 937, 1202, 1493, 1900, 2349, 2944, 3621, 4520, 5514, 6813, 8298, 10150, 12240, 14918, 17931, 21654, 25917, 31081, 37029, 44256, 52474, 62405, 73724, 87378, 102887
Offset: 1
The a(7) = 12 partitions:
(43), (52), (61),
(322), (331), (511),
(2221), (3211), (4111),
(22111), (31111),
(211111).
-
Table[Length[Select[IntegerPartitions[n],NestWhile[Sort[Length/@Split[#]]&,#,Length[#]>1&]=={2}&]],{n,30}]
Comments