A370348
Numbers k such that there are fewer divisors of prime indices of k than there are prime indices of k.
Original entry on oeis.org
4, 8, 12, 16, 18, 20, 24, 27, 32, 36, 40, 44, 48, 50, 54, 56, 60, 64, 68, 72, 80, 81, 84, 88, 90, 96, 100, 108, 112, 120, 124, 125, 126, 128, 132, 135, 136, 144, 150, 160, 162, 164, 168, 176, 180, 184, 189, 192, 196, 198, 200, 204, 208, 216, 220, 224, 225, 236, 240, 242, 243, 248, 250, 252, 256
Offset: 1
a(5) = 18 is a term because the prime indices of 18 = 2 * 3^2 are 1,2,2, and there are 3 of these but only 2 divisors of prime indices, namely 1 and 2.
For submultisets instead of parts on the RHS we get
A371167.
Partitions of this type are counted by
A371171.
A001221 counts distinct prime factors.
A355731 counts choices of a divisor of each prime index, firsts
A355732.
-
filter:= proc(n) uses numtheory; local F,D,t;
F:= map(t -> [pi(t[1]),t[2]], ifactors(n)[2]);
D:= `union`(seq(divisors(t[1]), t = F));
nops(D) < add(t[2], t = F)
end proc:
select(filter, [$1..300]);
-
filter[n_] := Module[{F, d},
F = {PrimePi[#[[1]]], #[[2]]}& /@ FactorInteger[n];
d = Union[Flatten[Divisors /@ F[[All, 1]]]];
Length[d] < Total[F[[All, 2]]]];
Select[Range[300], filter] (* Jean-François Alcover, Mar 08 2024, after Maple code *)
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}]
A371128
Number of strict integer partitions of n containing all distinct divisors of all parts.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 2, 1, 2, 1, 2, 2, 3, 3, 3, 5, 3, 5, 6, 7, 7, 8, 8, 9, 12, 13, 13, 14, 15, 16, 19, 23, 25, 26, 26, 27, 36, 37, 40, 42, 46, 50, 55, 66, 65, 71, 71, 82, 90, 102, 103, 114, 117, 130, 147, 154, 166, 176, 182, 194, 228, 239, 259, 267, 287, 307, 336
Offset: 0
The a(9) = 1 through a(19) = 7 partitions (A..H = 10..17):
531 721 731 B1 751 D1 B31 D21 B51 H1 B71
4321 5321 5421 931 B21 7521 7531 D31 9531 D51
6321 7321 7421 8421 64321 B321 A521 B521
9321 65321 B421 D321
54321 74321 75321 75421
84321 76321
94321
A008284 counts partitions by length.
-
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&SubsetQ[#,Union@@Divisors/@#]&]],{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}]
A371165
Positive integers with as many divisors (A000005) as distinct divisors of prime indices (A370820).
Original entry on oeis.org
3, 5, 11, 17, 26, 31, 35, 38, 39, 41, 49, 57, 58, 59, 65, 67, 69, 77, 83, 86, 87, 94, 109, 119, 127, 129, 133, 146, 148, 157, 158, 179, 191, 202, 206, 211, 217, 235, 237, 241, 244, 253, 274, 277, 278, 283, 284, 287, 291, 298, 303, 319, 326, 331, 333, 334, 353
Offset: 1
The terms together with their prime indices begin:
3: {2} 67: {19} 158: {1,22}
5: {3} 69: {2,9} 179: {41}
11: {5} 77: {4,5} 191: {43}
17: {7} 83: {23} 202: {1,26}
26: {1,6} 86: {1,14} 206: {1,27}
31: {11} 87: {2,10} 211: {47}
35: {3,4} 94: {1,15} 217: {4,11}
38: {1,8} 109: {29} 235: {3,15}
39: {2,6} 119: {4,7} 237: {2,22}
41: {13} 127: {31} 241: {53}
49: {4,4} 129: {2,14} 244: {1,1,18}
57: {2,8} 133: {4,8} 253: {5,9}
58: {1,10} 146: {1,21} 274: {1,33}
59: {17} 148: {1,1,12} 277: {59}
65: {3,6} 157: {37} 278: {1,34}
For prime factors instead of divisors on both sides we get
A319899.
For prime factors on LHS we get
A370802, for distinct prime factors
A371177.
For (greater than) instead of (equal) we get
A371166.
For (less than) instead of (equal) we get
A371167.
Partitions of this type are counted by
A371172.
A001221 counts distinct prime factors.
A355731 counts choices of a divisor of each prime index, firsts
A355732.
-
Select[Range[100],Length[Divisors[#]] == Length[Union@@Divisors/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]]&]
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.
A371178
Number of integer partitions of n containing all divisors of all parts.
Original entry on oeis.org
1, 1, 1, 2, 3, 4, 6, 9, 12, 16, 21, 28, 37, 48, 62, 80, 101, 127, 162, 202, 252, 312, 386, 475, 585, 713, 869, 1056, 1278, 1541, 1859, 2232, 2675, 3196, 3811, 4534, 5386, 6379, 7547, 8908, 10497, 12345, 14501, 16999, 19897, 23253, 27135, 31618, 36796, 42756
Offset: 0
The partition (4,2,1,1) contains all distinct divisors {1,2,4}, so is counted under a(8).
The partition (4,4,3,2,2,2,1) contains all distinct divisors {1,2,3,4} so is counted under 4 + 4 + 3 + 2 + 2 + 2 + 1 = 18. - _David A. Corneth_, Mar 18 2024
The a(0) = 1 through a(8) = 12 partitions:
() (1) (11) (21) (31) (221) (51) (331) (71)
(111) (211) (311) (321) (421) (521)
(1111) (2111) (2211) (511) (3221)
(11111) (3111) (2221) (3311)
(21111) (3211) (4211)
(111111) (22111) (5111)
(31111) (22211)
(211111) (32111)
(1111111) (221111)
(311111)
(2111111)
(11111111)
For partitions with no divisors of parts we have
A305148, ranks
A316476.
The complement is counted by
A371132.
For submultisets instead of distinct parts we have
A371172, ranks
A371165.
These partitions have ranks
A371177.
A008284 counts partitions by length.
Cf.
A000837,
A003963,
A239312,
A285573,
A305148,
A319055,
A355529,
A370803,
A370808,
A370813,
A371168,
A371171,
A371173.
-
Table[Length[Select[IntegerPartitions[n],SubsetQ[#,Union@@Divisors/@#]&]],{n,0,30}]
A371170
Positive integers with at most as many prime factors (A001222) as distinct divisors of prime indices (A370820).
Original entry on oeis.org
1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 92
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:
1: {} 22: {1,5} 42: {1,2,4} 63: {2,2,4}
2: {1} 23: {9} 43: {14} 65: {3,6}
3: {2} 25: {3,3} 45: {2,2,3} 66: {1,2,5}
5: {3} 26: {1,6} 46: {1,9} 67: {19}
6: {1,2} 28: {1,1,4} 47: {15} 69: {2,9}
7: {4} 29: {10} 49: {4,4} 70: {1,3,4}
9: {2,2} 30: {1,2,3} 51: {2,7} 71: {20}
10: {1,3} 31: {11} 52: {1,1,6} 73: {21}
11: {5} 33: {2,5} 53: {16} 74: {1,12}
13: {6} 34: {1,7} 55: {3,5} 75: {2,3,3}
14: {1,4} 35: {3,4} 57: {2,8} 76: {1,1,8}
15: {2,3} 37: {12} 58: {1,10} 77: {4,5}
17: {7} 38: {1,8} 59: {17} 78: {1,2,6}
19: {8} 39: {2,6} 61: {18} 79: {22}
21: {2,4} 41: {13} 62: {1,11} 82: {1,13}
A001221 counts distinct prime factors.
A355731 counts choices of a divisor of each prime index, firsts
A355732.
-
Select[Range[100],PrimeOmega[#]<=Length[Union @@ Divisors/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]]&]
A371169
Positive integers with at least as many prime factors (A001222) as distinct divisors of prime indices (A370820).
Original entry on oeis.org
1, 2, 4, 6, 8, 9, 10, 12, 16, 18, 20, 22, 24, 25, 27, 28, 30, 32, 34, 36, 40, 42, 44, 45, 48, 50, 54, 56, 60, 62, 63, 64, 66, 68, 72, 75, 80, 81, 82, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 108, 110, 112, 118, 120, 121, 124, 125, 126, 128, 132, 134, 135
Offset: 1
The terms together with their prime indices begin:
1: {}
2: {1}
4: {1,1}
6: {1,2}
8: {1,1,1}
9: {2,2}
10: {1,3}
12: {1,1,2}
16: {1,1,1,1}
18: {1,2,2}
20: {1,1,3}
22: {1,5}
24: {1,1,1,2}
25: {3,3}
27: {2,2,2}
28: {1,1,4}
30: {1,2,3}
32: {1,1,1,1,1}
34: {1,7}
36: {1,1,2,2}
The version for prime factors instead of divisors on the RHS is
A319899.
A001221 counts distinct prime factors.
A355731 counts choices of a divisor of each prime index, firsts
A355732.
-
Select[Range[100],PrimeOmega[#]>=Length[Union @@ Divisors/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]]&]
A371166
Positive integers with fewer divisors (A000005) than distinct divisors of prime indices (A370820).
Original entry on oeis.org
7, 13, 19, 23, 29, 37, 43, 47, 53, 61, 71, 73, 74, 79, 89, 91, 95, 97, 101, 103, 106, 107, 111, 113, 122, 131, 137, 139, 141, 142, 143, 145, 149, 151, 159, 161, 163, 167, 169, 173, 178, 181, 183, 185, 193, 197, 199, 203, 209, 213, 214, 215, 219, 221, 223, 226
Offset: 1
The terms together with their prime indices begin:
7: {4} 101: {26} 163: {38} 223: {48}
13: {6} 103: {27} 167: {39} 226: {1,30}
19: {8} 106: {1,16} 169: {6,6} 227: {49}
23: {9} 107: {28} 173: {40} 229: {50}
29: {10} 111: {2,12} 178: {1,24} 233: {51}
37: {12} 113: {30} 181: {42} 239: {52}
43: {14} 122: {1,18} 183: {2,18} 247: {6,8}
47: {15} 131: {32} 185: {3,12} 251: {54}
53: {16} 137: {33} 193: {44} 257: {55}
61: {18} 139: {34} 197: {45} 259: {4,12}
71: {20} 141: {2,15} 199: {46} 262: {1,32}
73: {21} 142: {1,20} 203: {4,10} 263: {56}
74: {1,12} 143: {5,6} 209: {5,8} 265: {3,16}
79: {22} 145: {3,10} 213: {2,20} 267: {2,24}
89: {24} 149: {35} 214: {1,28} 269: {57}
91: {4,6} 151: {36} 215: {3,14} 271: {58}
95: {3,8} 159: {2,16} 219: {2,21} 281: {60}
97: {25} 161: {4,9} 221: {6,7} 293: {62}
For (equal to) instead of (less than) we have
A371165, counted by
A371172.
For (greater than) instead of (less than) we have
A371167.
A001221 counts distinct prime factors.
A355731 counts choices of a divisor of each prime index, firsts
A355732.
Showing 1-10 of 14 results.
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