cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A370585 Number of maximal subsets of {1..n} such that it is possible to choose a different prime factor of each element.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 5, 5, 7, 11, 25, 25, 38, 38, 84, 150, 178, 178, 235, 235, 341, 579, 1235, 1235, 1523, 1968, 4160, 4824, 6840, 6840, 9140, 9140, 10028, 16264, 33956, 48680, 56000, 56000, 116472, 186724, 223884, 223884, 290312, 290312, 403484, 484028, 1001420
Offset: 0

Views

Author

Gus Wiseman, Feb 26 2024

Keywords

Comments

First differs from A307984 at a(21) = 579, A307984(21) = 578. The difference is due to the set {10,11,13,14,15,17,19,21}, which is not a basis because log(10) + log(21) = log(14) + log(15).
Also length-pi(n) subsets of {1..n} such that it is possible to choose a different prime factor of each element.

Examples

			The a(0) = 1 through a(8) = 7 subsets:
  {}  {}  {2}  {2,3}  {2,3}  {2,3,5}  {2,3,5}  {2,3,5,7}  {2,3,5,7}
                      {3,4}  {3,4,5}  {2,5,6}  {2,5,6,7}  {2,5,6,7}
                                      {3,4,5}  {3,4,5,7}  {3,4,5,7}
                                      {3,5,6}  {3,5,6,7}  {3,5,6,7}
                                      {4,5,6}  {4,5,6,7}  {3,5,7,8}
                                                          {4,5,6,7}
                                                          {5,6,7,8}

Crossrefs

Multisets of this type are ranked by A368100, complement A355529.
Factorizations of this type are counted by A368414, complement A368413.
The version for set-systems is A368601, max of A367902 (complement A367903).
This is the maximal case of A370582, complement A370583, cf. A370584.
A different kind of maximality is A370586, complement A370587.
The case containing n is A370590, complement A370591.
Partitions of this type (choosable) are A370592, complement A370593.
For binary indices instead of factors we have A370640, cf. A370636, A370637.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A307984 counts Q-bases of logarithms of positive integers.
A355741 counts choices of a prime factor of each prime index.
Cf. A000040, A000720, A005117, A045778, A133686, A333331, A355739, A355740, A355744, A355745, A367905, A368110.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n], {PrimePi[n]}],Length[Select[Tuples[If[#==1, {},First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]>0&]],{n,0,10}]

Extensions

More terms from Jinyuan Wang, Feb 14 2025

A370582 Number of subsets of {1..n} such that it is possible to choose a different prime factor of each element.

Original entry on oeis.org

1, 1, 2, 4, 6, 12, 20, 40, 52, 72, 116, 232, 320, 640, 1020, 1528, 1792, 3584, 4552, 9104, 12240, 17840, 27896, 55792, 67584, 83968, 130656, 150240, 198528, 397056, 507984, 1015968, 1115616, 1579168, 2438544, 3259680, 3730368, 7460736, 11494656, 16145952, 19078464, 38156928
Offset: 0

Views

Author

Gus Wiseman, Feb 25 2024

Keywords

Examples

			The a(0) = 1 through a(6) = 20 subsets:
  {}  {}  {}   {}     {}     {}       {}
          {2}  {2}    {2}    {2}      {2}
               {3}    {3}    {3}      {3}
               {2,3}  {4}    {4}      {4}
                      {2,3}  {5}      {5}
                      {3,4}  {2,3}    {6}
                             {2,5}    {2,3}
                             {3,4}    {2,5}
                             {3,5}    {2,6}
                             {4,5}    {3,4}
                             {2,3,5}  {3,5}
                             {3,4,5}  {3,6}
                                      {4,5}
                                      {4,6}
                                      {5,6}
                                      {2,3,5}
                                      {2,5,6}
                                      {3,4,5}
                                      {3,5,6}
                                      {4,5,6}

Crossrefs

The version for set-systems is A367902, ranks A367906, unlabeled A368095.
The complement for set-systems is A367903, ranks A367907, unlabeled A368094.
For unlabeled multiset partitions we have A368098, complement A368097.
Multisets of this type are ranked by A368100, complement A355529.
For divisors instead of factors we have A368110, complement A355740.
The version for factorizations is A368414, complement A368413.
The complement is counted by A370583.
For a unique choice we have A370584.
The maximal case is A370585.
Partial sums of A370586, complement A370587.
The version for partitions is A370592, complement A370593.
For binary indices instead of factors we have A370636, complement A370637.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A307984 counts Q-bases of logarithms of positive integers.
A355741 counts choices of a prime factor of each prime index.
Cf. A000040, A000720, A001055, A001414, A003963, A005117, A045778, A133686, A355739, A355744, A355745, A367905.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],Length[Select[Tuples[If[#==1,{},First/@FactorInteger[#]]&/@#],UnsameQ@@#&]]>0&]],{n,0,10}]

Formula

a(p) = 2 * a(p-1) for prime p. - David A. Corneth, Feb 25 2024
a(n) = 2^n - A370583(n).

Extensions

a(19) from David A. Corneth, Feb 25 2024
a(20)-a(41) from Alois P. Heinz, Feb 25 2024

A370802 Positive integers with as many prime factors (A001222) as distinct divisors of prime indices (A370820).

Original entry on oeis.org

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
Offset: 1

Views

Author

Gus Wiseman, Mar 14 2024

Keywords

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.

Examples

			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}

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.
The RHS is A370820, for prime factors instead of divisors A303975.
Partitions of this type are counted by A371130, strict A371128.
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.
Other inequalities: A371166, A371167, A371169, A371170.
A000005 counts divisors.
A001221 counts distinct prime factors.
A027746 lists prime factors, A112798 indices, length A001222.
A239312 counts divisor-choosable partitions, ranks A368110.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A370320 counts non-divisor-choosable partitions, ranks A355740.
Cf. A000792, A003963, A355529, A355737, A355739, A355741, A368100, A370808, A370813, A370814, A371127.

Programs

  • Mathematica
    Select[Range[100],PrimeOmega[#]==Length[Union @@ Divisors/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]]&]

Formula

A001222(a(n)) = A370820(a(n)).

A370584 Number of subsets of {1..n} such that only one set can be obtained by choosing a different prime factor of each element.

Original entry on oeis.org

1, 1, 2, 4, 6, 12, 18, 36, 48, 68, 104, 208, 284, 568, 888, 1296, 1548, 3096, 3968, 7936, 10736, 15440, 24008, 48016, 58848, 73680, 114368, 132608, 176240, 352480, 449824, 899648, 994976, 1399968, 2160720, 2859584, 3296048, 6592096, 10156672, 14214576, 16892352
Offset: 0

Views

Author

Gus Wiseman, Feb 26 2024

Keywords

Comments

For example, the only choice of a different prime factor of each element of (4,5,6) is (2,5,3).

Examples

			The a(0) = 1 through a(6) = 18 subsets:
  {}  {}  {}   {}     {}     {}       {}
          {2}  {2}    {2}    {2}      {2}
               {3}    {3}    {3}      {3}
               {2,3}  {4}    {4}      {4}
                      {2,3}  {5}      {5}
                      {3,4}  {2,3}    {2,3}
                             {2,5}    {2,5}
                             {3,4}    {2,6}
                             {3,5}    {3,4}
                             {4,5}    {3,5}
                             {2,3,5}  {3,6}
                             {3,4,5}  {4,5}
                                      {4,6}
                                      {2,3,5}
                                      {2,5,6}
                                      {3,4,5}
                                      {3,5,6}
                                      {4,5,6}

Crossrefs

For divisors instead of factors we have A051026, cf. A368110, A355740.
The version for set-systems is A367904, ranks A367908.
Multisets of this type are ranked by A368101, cf. A368100, A355529.
For existence we have A370582, differences A370586.
For nonexistence we have A370583, differences A370587.
Maximal sets of this type are counted by A370585.
The version for partitions is A370594, cf. A370592, A370593.
For binary indices instead of factors we have A370638, cf. A370636, A370637.
The version for factorizations is A370645, cf. A368414, A368413.
For unlabeled multiset partitions we have A370646, cf. A368098, A368097.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A355741 counts ways to choose a prime factor of each prime index.
Cf. A000040, A000720, A003963, A005117, A045778, A133686, A307984, A355739, A355744, A355745, A367905.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]], Length[Union[Sort/@Select[Tuples[If[#==1, {},First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]]==1&]],{n,0,10}]

Extensions

More terms from Jinyuan Wang, Mar 28 2025

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

Views

Author

Robert Israel, Feb 15 2024

Keywords

Comments

No multiple of a term is a term of A368110.

Examples

			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.

Links

Crossrefs

The LHS is A370820, firsts A371131.
The version for equality is A370802, counted by A371130, strict A371128.
For submultisets instead of parts on the RHS we get A371167.
The opposite version is A371168, counted by A371173.
The weak version is A371169.
The complement is A371170.
Partitions of this type are counted by A371171.
A000005 counts divisors.
A001221 counts distinct prime factors.
A027746 lists prime factors, indices A112798, length A001222.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Cf. A003963, A239312, A303975, A319899, A355529, A355739, A370808, A371127.

Programs

  • Maple
    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]);
  • Mathematica
    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 *)

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

Views

Author

Gus Wiseman, Mar 03 2024

Keywords

Examples

			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

Crossrefs

Including partitions with one choice gives A239312, complement A370320.
For a unique choice we have A370595, ranks A370810.
These partitions have ranks A370811.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A027746 lists prime factors, A112798 indices, length A001222.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355733 counts divisor-choices of prime indices.
A355741, A355744, A355745 choose prime factors of prime indices.
A370592 counts factor-choosable partitions, ranks A368100.
A370593 counts non-factor-choosable partitions, ranks A355529.
Cf. A355739, A368110, A370594, A370804, A370805, A370808, A370814.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Length[Union[Sort /@ Select[Tuples[Divisors/@#],UnsameQ@@#&]]]>1&]],{n,0,30}]

Formula

a(n) = A239312(n) - A370595(n). - Jinyuan Wang, Feb 14 2025

Extensions

More terms from Jinyuan Wang, Feb 14 2025

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

Views

Author

Gus Wiseman, Mar 17 2024

Keywords

Comments

The Heinz numbers of these partitions are given by A370802.

Examples

			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)

Crossrefs

The LHS is represented by A001222, distinct A000021.
These partitions are ranked by A370802.
The RHS is represented by A370820, for prime factors A303975.
The strict case is A371128.
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.
Counting only distinct parts on the LHS gives A371178, ranks A371177.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length.
Choosable partitions: A239312 (A368110), A355740 (A370320), A370592 (A368100), A370593 (A355529).
Cf. A003963, A319055, A355731, A370803, A370808, A370809, A370813, A370814.

Programs

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

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

Views

Author

Gus Wiseman, Feb 29 2024

Keywords

Examples

			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)

Crossrefs

The version for set-systems is A367904, ranks A367908.
Multisets of this type are ranked by A368101, cf. A368100, A355529.
The version for subsets is A370584, cf. A370582, A370583, A370586, A370587.
Maximal sets of this type are counted by A370585.
For existence we have A370592.
For nonexistence we have A370593.
For divisors instead of factors we have A370595.
For subsets and binary indices we have A370638, cf. A370636, A370637.
The version for factorizations is A370645, cf. A368414, A368413.
These partitions have ranks A370647.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A355741 counts ways to choose a prime factor of each prime index.
Cf. A000040, A000041, A000720, A003963, A355739, A355740, A367905, A368110.

Programs

  • Mathematica
    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

Views

Author

Gus Wiseman, Mar 16 2024

Keywords

Comments

The Heinz numbers of these partitions are given by A370348.

Examples

			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)

Crossrefs

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 ranks A371167.
A000005 counts divisors.
Choosable partitions: A239312 (A368110), A355740 (A370320), A370592 (A368100), A370593 (A355529).
Cf. A003963, A319055, A355731, A370803, A370808, A370809, A370813, A370814.

Programs

  • Mathematica
    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

Views

Author

Gus Wiseman, Mar 03 2024

Keywords

Comments

For example, the only choice for the partition (9,9,6,6,6) is {1,2,3,6,9}.

Examples

			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

Crossrefs

For no choices we have A370320, complement A239312.
The version for prime factors (not all divisors) is A370594, ranks A370647.
For multiple choices we have A370803, ranks A370811.
These partitions have ranks A370810.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A027746 lists prime factors, A112798 indices, length A001222.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355741, A355744, A355745 choose prime factors of prime indices.
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.
Cf. A355535, A355739, A355740, A367867, A367904, A368110, A370583, A370584, A370806, A370808.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Length[Union[Sort /@ Select[Tuples[Divisors/@#],UnsameQ@@#&]]]==1&]],{n,0,30}]

Extensions

More terms from Jinyuan Wang, Feb 14 2025
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