Displaying 1-10 of 31 results found.
Number of condensed integer partitions of n.
+10
53
1, 1, 1, 2, 3, 3, 5, 6, 9, 10, 14, 16, 23, 27, 33, 41, 51, 62, 75, 93, 111, 134, 159, 189, 226, 271, 317, 376, 445, 520, 609, 714, 832, 972, 1129, 1304, 1520, 1753, 2023, 2326, 2692, 3077, 3540, 4050, 4642, 5298, 6054, 6887, 7854, 8926, 10133, 11501, 13044
COMMENTS
Suppose that p is a partition of n. Let x(1), x(2), ..., x(k) be the distinct parts of p, and let m(i) be the multiplicity of x(i) in p. Let c(p) be the partition {m(1)*x(1), m(2)*x(2), ..., x(k)*m(k)} of n. Call a partition q of n a condensed partition of n if q = c(p) for some partition p of n. Then a(n) is the number of distinct condensed partitions of n. Note that c(p) = p if and only if p has distinct parts and that condensed partitions can have repeated parts.
Also the number of integer partitions of n such that it is possible to choose a different divisor of each part. For example, the partition (6,4,4,1) has choices (3,2,4,1), (3,4,2,1), (6,2,4,1), (6,4,2,1) so is counted under a(15). - Gus Wiseman, Mar 12 2024
EXAMPLE
a(5) = 3 gives the number of partitions of 5 that result from condensations as shown here: 5 -> 5, 41 -> 41, 32 -> 32, 311 -> 32, 221 -> 41, 2111 -> 32, 11111 -> 5.
The a(1) = 1 through a(9) = 10 condensed partitions:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(2,1) (2,2) (3,2) (3,3) (4,3) (4,4) (5,4)
(3,1) (4,1) (4,2) (5,2) (5,3) (6,3)
(5,1) (6,1) (6,2) (7,2)
(3,2,1) (3,2,2) (7,1) (8,1)
(4,2,1) (3,3,2) (4,3,2)
(4,2,2) (4,4,1)
(4,3,1) (5,2,2)
(5,2,1) (5,3,1)
(6,2,1)
(End)
MAPLE
b:= proc(n, i) option remember; `if`(n=0, {[]},
`if`(i=1, {[n]}, {seq(map(x-> `if`(j=0, x,
sort([x[], i*j])), b(n-i*j, i-1))[], j=0..n/i)}))
end:
a:= n-> nops(b(n$2)):
MATHEMATICA
u[n_, k_] := u[n, k] = Map[Total, Split[IntegerPartitions[n][[k]]]]; t[n_] := t[n] = DeleteDuplicates[Table[Sort[u[n, k]], {k, 1, PartitionsP[n]}]]; Table[Length[t[n]], {n, 0, 30}]
Table[Length[Select[IntegerPartitions[n], Length[Select[Tuples[Divisors/@#], UnsameQ@@#&]]>0&]], {n, 0, 30}] (* Gus Wiseman, Mar 12 2024 *)
CROSSREFS
The complement is counted by A370320. The version for prime factors (not all divisors) is A370592, ranks A368100. A237685 counts partitions of depth 1, or A353837 if we include depth 0.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Cf. A355535, A355733, A355739, A367867, A368097, A368414, A370583, A370584, A370594, A370806, A370808.
Number of factorizations of n into positive integers > 1 such that it is not possible to choose a different prime factor of each factor.
+10
41
0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 4, 0, 1, 0, 1, 0, 0, 0, 3, 1, 0, 2, 1, 0, 0, 0, 6, 0, 0, 0, 4, 0, 0, 0, 3, 0, 0, 0, 1, 1, 0, 0, 7, 1, 1, 0, 1, 0, 3, 0, 3, 0, 0, 0, 2, 0, 0, 1, 10, 0, 0, 0, 1, 0, 0, 0, 10, 0, 0, 1, 1, 0, 0, 0, 7, 4, 0, 0, 2, 0, 0
COMMENTS
For example, the factorization f = 2*3*6 has two ways to choose a prime factor of each factor, namely (2,3,2) and (2,3,3), but neither of these has all different elements, so f is counted under a(36).
EXAMPLE
The a(1) = 0 through a(24) = 3 factorizations:
... 2*2 ... 2*4 3*3 .. 2*2*3 ... 2*8 . 2*3*3 . 2*2*5 ... 2*2*6
2*2*2 4*4 2*3*4
2*2*4 2*2*2*3
2*2*2*2
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], Select[Tuples[First/@FactorInteger[#]&/@#], UnsameQ@@#&]=={}&]], {n, 100}]
CROSSREFS
The complement is counted by A368414.
Number of factorizations of n into positive integers > 1 such that it is possible to choose a different prime factor of each factor.
+10
37
1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 5, 1, 1, 2, 2, 2, 5, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 9, 1, 2, 3, 1, 2, 5, 1, 3, 2, 5, 1, 6, 1, 2, 3, 3, 2, 5, 1, 5, 1, 2, 1, 9, 2, 2, 2
COMMENTS
For example, the factorization f = 2*3*6 has two ways to choose a prime factor of each factor, namely (2,3,2) and (2,3,3), but neither of these has all different elements, so f is not counted under a(36).
EXAMPLE
The a(n) factorizations for selected n:
1 6 12 24 30 60 72 120
2*3 2*6 2*12 2*15 2*30 2*36 2*60
3*4 3*8 3*10 3*20 3*24 3*40
4*6 5*6 4*15 4*18 4*30
2*3*5 5*12 6*12 5*24
6*10 8*9 6*20
2*3*10 8*15
2*5*6 10*12
3*4*5 2*3*20
2*5*12
2*6*10
3*4*10
3*5*8
4*5*6
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join @@ Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], Select[Tuples[First/@FactorInteger[#]&/@#], UnsameQ@@#&]!={}&]], {n, 100}]
CROSSREFS
The complement is counted by A368413.
Number of condensed integer factorizations of n into unordered factors > 1.
+10
29
1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 2, 4, 1, 5, 1, 5, 2, 2, 2, 8, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 10, 2, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 11, 1, 2, 4, 7, 2, 5, 1, 4, 2, 5, 1, 14, 1, 2, 4, 4, 2, 5, 1, 10, 4, 2, 1, 11, 2
COMMENTS
A multiset is condensed iff it is possible to choose a different divisor of each element.
EXAMPLE
The a(36) = 7 factorizations: (2*2*9), (2*3*6), (2*18), (3*3*4), (3*12), (4*9), (6*6), (36).
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join @@ Table[Map[Prepend[#, d]&, Select[facs[n/d], Min @@ #>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], Length[Select[Tuples[Divisors /@ #], UnsameQ@@#&]]>0&]], {n, 100}]
CROSSREFS
A355731 counts choices of a divisor of each prime index, firsts A355732.
Number of subsets of {1..n} such that it is possible to choose a different binary index of each element.
+10
27
1, 2, 4, 7, 14, 24, 39, 61, 122, 203, 315, 469, 676, 952, 1307, 1771, 3542, 5708, 8432, 11877, 16123, 21415, 27835, 35757, 45343, 57010, 70778, 87384, 106479, 129304, 155802, 187223, 374446, 588130, 835800, 1124981, 1456282, 1841361, 2281772, 2791896, 3367162
COMMENTS
A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
EXAMPLE
The a(0) = 1 through a(4) = 14 subsets:
{} {} {} {} {}
{1} {1} {1} {1}
{2} {2} {2}
{1,2} {3} {3}
{1,2} {4}
{1,3} {1,2}
{2,3} {1,3}
{1,4}
{2,3}
{2,4}
{3,4}
{1,2,4}
{1,3,4}
{2,3,4}
MATHEMATICA
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Table[Length[Select[Subsets[Range[n]], Select[Tuples[bpe/@#], UnsameQ@@#&]!={}&]], {n, 0, 10}]
CROSSREFS
Unlabeled graphs of this type are counted by A134964, complement A140637. Simple graphs not of this type are counted by A367867, covering A367868. Set systems uniquely of this type are counted by A367904, ranks A367908. Unlabeled multiset partitions of this type are A368098, complement A368097. A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326031 gives weight of the set-system with BII-number n.
Number of non-condensed integer partitions of n, or partitions where it is not possible to choose a different divisor of each part.
+10
25
0, 0, 1, 1, 2, 4, 6, 9, 13, 20, 28, 40, 54, 74, 102, 135, 180, 235, 310, 397, 516, 658, 843, 1066, 1349, 1687, 2119, 2634, 3273, 4045, 4995, 6128, 7517, 9171, 11181, 13579, 16457, 19884, 23992, 28859, 34646, 41506, 49634, 59211, 70533, 83836, 99504, 117867
COMMENTS
Includes all partitions containing 1.
EXAMPLE
The a(0) = 0 through a(8) = 13 partitions:
. . (11) (111) (211) (221) (222) (331) (611)
(1111) (311) (411) (511) (2222)
(2111) (2211) (2221) (3221)
(11111) (3111) (3211) (3311)
(21111) (4111) (4211)
(111111) (22111) (5111)
(31111) (22211)
(211111) (32111)
(1111111) (41111)
(221111)
(311111)
(2111111)
(11111111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Length[Select[Tuples[Divisors/@#], UnsameQ@@#&]]==0&]], {n, 0, 30}]
CROSSREFS
The complement is counted by A239312 (condensed partitions). These partitions have ranks A355740. Factorizations in the case of prime factors are A368413, complement A368414. The version for prime factors (not all divisors) is A370593, ranks A355529. A355731 counts choices of a divisor of each prime index, firsts A355732.
Cf. A355535, A355739, A367867, A368097, A368110, A370583, A370584, A370594, A370806, A370807, A370808.
Positive integers with as many prime factors ( A001222) as distinct divisors of prime indices ( A370820).
+10
22
1, 2, 6, 9, 10, 22, 25, 28, 30, 34, 42, 45, 62, 63, 66, 75, 82, 92, 98, 99, 102, 104, 110, 118, 121, 134, 140, 147, 152, 153, 156, 166, 170, 186, 210, 218, 228, 230, 232, 234, 246, 254, 260, 275, 276, 279, 289, 308, 310, 314, 315, 330, 342, 343, 344, 348, 350
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. All squarefree terms are even.
EXAMPLE
The prime indices of 1617 are {2,4,4,5}, with distinct divisors {1,2,4,5}, so 1617 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}
28: {1,1,4}
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}
92: {1,1,9}
98: {1,4,4}
99: {2,2,5}
102: {1,2,7}
104: {1,1,1,6}
MATHEMATICA
Select[Range[100], PrimeOmega[#]==Length[Union @@ Divisors/@PrimePi/@First/@If[#==1, {}, FactorInteger[#]]]&]
CROSSREFS
For factors instead of divisors on the RHS we have A319899. A version for binary indices is A367917. For (greater than) instead of (equal) we have A370348, counted by A371171. For divisors instead of factors on LHS we have A371165, counted by A371172. For only distinct prime factors on LHS we have A371177, counted by A371178. A001221 counts distinct prime factors.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Cf. A000792, A003963, A355529, A355737, A355739, A355741, A368100, A370808, A370813, A370814, A371127.
Number of subsets of {1..n} such that it is not possible to choose a different binary index of each element.
+10
20
0, 0, 0, 1, 2, 8, 25, 67, 134, 309, 709, 1579, 3420, 7240, 15077, 30997, 61994, 125364, 253712, 512411, 1032453, 2075737, 4166469, 8352851, 16731873, 33497422, 67038086, 134130344, 268328977, 536741608, 1073586022, 2147296425, 4294592850, 8589346462, 17179033384
COMMENTS
A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
EXAMPLE
The a(0) = 0 through a(5) = 8 subsets:
. . . {1,2,3} {1,2,3} {1,2,3}
{1,2,3,4} {1,4,5}
{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}
MATHEMATICA
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Table[Length[Select[Subsets[Range[n]], Select[Tuples[bpe/@#], UnsameQ@@#&]=={}&]], {n, 0, 10}]
CROSSREFS
Simple graphs not of this type are counted by A133686, covering A367869. Unlabeled graphs of this type are counted by A140637, complement A134964. Set systems uniquely not of this type are counted by A367904, ranks A367908. Unlabeled multiset partitions of this type are A368097, complement A368098. A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326031 gives weight of the set-system with BII-number n.
Cf. A000612, A072639, A355739, A355740, A367772, A367905, A367909, A367912, A368094, A368095, A368109.
Number of integer partitions of n such that the number of parts is equal to the number of distinct divisors of parts.
+10
18
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
COMMENTS
The Heinz numbers of these partitions are given by A370802.
EXAMPLE
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)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Length[#]==Length[Union@@Divisors/@#]&]], {n, 0, 30}]
CROSSREFS
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.
Number of integer partitions of n with more parts than distinct divisors of parts.
+10
15
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
COMMENTS
The Heinz numbers of these partitions are given by A370348.
EXAMPLE
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)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Length[#] > Length[Union@@Divisors/@#]&]], {n, 0, 30}]
CROSSREFS
The partitions are ranked by A370348. For submultisets instead of parts on the LHS we get ranks A371167.
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