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.

Previous Showing 11-20 of 31 results. Next

A371177 Positive integers whose prime indices include all distinct divisors of all prime indices.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 30, 32, 34, 36, 40, 42, 44, 48, 50, 54, 60, 62, 64, 66, 68, 72, 80, 82, 84, 88, 90, 96, 100, 102, 108, 110, 118, 120, 124, 126, 128, 132, 134, 136, 144, 150, 160, 162, 164, 166, 168, 170, 176, 180, 186, 192, 198, 200
Offset: 1

Views

Author

Gus Wiseman, Mar 18 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.
Also positive integers with as many distinct prime factors (A001221) as distinct divisors of prime indices (A370820).

Examples

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   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}
   30: {1,2,3}
   32: {1,1,1,1,1}
   34: {1,7}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   48: {1,1,1,1,2}

Crossrefs

The LHS is A001221, distinct case of A001222.
The RHS is A370820, for prime factors A303975.
For bigomega on the LHS we have A370802, counted by A371130.
For divisors on the LHS we have A371165, counted by A371172.
Partitions of this type are counted by A371178, strict A371128.
The complement is A371179, counted by A371132.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length.
A305148 counts partitions without divisors, strict A303362, ranks A316476.
Cf. A000837, A003963, A239312, A285573, A355529, A370813, A371168.

Programs

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

Formula

A001221(a(n)) = A370820(a(n)).

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

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

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.

Examples

			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}

Crossrefs

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.
The RHS is A370820, for prime factors instead of divisors A303975.
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.
Other inequalities: A370348 (A371171), A371168 (A371173), 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.
A370814 counts divisor-choosable factorizations, complement A370813.
Cf. A000792, A003963, A355529, A355739, A355741, A368100, A370808, A371127.

Programs

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

Formula

A000005(a(n)) = A370820(a(n)).

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

Views

Author

Gus Wiseman, Mar 16 2024

Keywords

Comments

The Heinz numbers of these partitions are given by A371168.

Examples

			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)

Crossrefs

The RHS is represented by A370820.
The version for equality is A371130 (ranks A370802), strict A371128.
For submultisets instead of parts on the LHS we get ranks A371166.
These partitions are ranked by A371168.
The opposite version is A371171, ranks A370348.
A000005 counts divisors.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Choosable partitions: A239312 (A368110), A355740 (A370320), A370592 (A368100), A370593 (A355529).
Cf. A003963, A319055, A355739, A370803, A370808, A370809, A370813, A370814.

Programs

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

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

Views

Author

Gus Wiseman, Mar 05 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.

Examples

			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}

Crossrefs

For no choices we have A355740, counted by A370320.
For at least one choice we have A368110, counted by A239312.
Partitions of this type are counted by A370595 and A370815.
For just prime factors we have A370647, counted by A370594.
For more than one choice we have A370811, counted by A370803.
A000005 counts divisors.
A006530 gives greatest prime factor, least A020639.
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.
A370814 counts factorizations with choosable divisors, complement A370813.
Cf. A133686, A355529, A355739, A355749, A367771, A367904, A370584, A370592, A370808, A370816.

Programs

  • Mathematica
    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&]

A371172 Number of integer partitions of n with as many submultisets as distinct divisors of parts.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 0, 3, 2, 3, 1, 4, 2, 1, 2, 3, 4, 2, 4, 1, 5, 2, 7, 5, 9, 4, 9, 15, 18, 16, 24, 13, 17, 23, 23, 22, 34, 17, 30, 31, 36, 29, 43, 21, 30, 35, 44, 28, 47, 19, 44
Offset: 0

Views

Author

Gus Wiseman, Mar 16 2024

Keywords

Comments

The Heinz numbers of these partitions are given by A371165.

Examples

			The partition (8,6,6) has 6 submultisets {(8,6,6),(8,6),(6,6),(8),(6),()} and 6 distinct divisors of parts {1,2,3,4,6,8}, so is counted under a(20).
The a(17) = 2 through a(24) = 9 partitions:
  (17)    (9,9)     (19)  (11,9)    (14,7)  (13,9)    (23)       (21,3)
  (13,4)  (15,3)          (15,5)    (17,4)  (21,1)    (19,4)     (22,2)
          (6,6,6)         (8,6,6)           (8,8,6)   (22,1)     (8,8,8)
          (12,3,3)        (12,4,4)          (10,6,6)  (15,4,4)   (10,8,6)
                          (18,1,1)          (16,3,3)  (12,10,1)  (12,6,6)
                                            (18,2,2)             (12,7,5)
                                            (20,1,1)             (18,3,3)
                                                                 (20,2,2)
                                                                 (12,10,2)

Crossrefs

The RHS is represented by A370820.
Counting parts on the LHS gives A371130 (ranks A370802), strict A371128.
These partitions are ranked by A371165.
A000005 counts divisors.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Choosable partitions: A239312 (A368110), A355740 (A370320), A370592 (A368100), A370593 (A355529).
Cf. A003963, A319055, A355739, A370803, A370808, A370813, A370814, A371166.

Programs

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

A370639 Number of subsets of {1..n} containing n such that it is possible to choose a different binary index of each element.

Original entry on oeis.org

0, 1, 2, 3, 7, 10, 15, 22, 61, 81, 112, 154, 207, 276, 355, 464, 1771, 2166, 2724, 3445, 4246, 5292, 6420, 7922, 9586, 11667, 13768, 16606, 19095, 22825, 26498, 31421, 187223, 213684, 247670, 289181, 331301, 385079, 440411, 510124, 575266, 662625, 747521
Offset: 0

Views

Author

Gus Wiseman, Mar 08 2024

Keywords

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.

Examples

			The a(0) = 0 through a(6) = 15 subsets:
  .  {1}  {2}    {3}    {4}      {5}      {6}
          {1,2}  {1,3}  {1,4}    {1,5}    {1,6}
                 {2,3}  {2,4}    {2,5}    {2,6}
                        {3,4}    {3,5}    {3,6}
                        {1,2,4}  {4,5}    {4,6}
                        {1,3,4}  {1,2,5}  {5,6}
                        {2,3,4}  {1,3,5}  {1,2,6}
                                 {2,3,5}  {1,3,6}
                                 {2,4,5}  {1,4,6}
                                 {3,4,5}  {1,5,6}
                                          {2,3,6}
                                          {2,5,6}
                                          {3,4,6}
                                          {3,5,6}
                                          {4,5,6}

Links

Crossrefs

Simple graphs of this type are counted by A133686, covering A367869.
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 of this type are counted by A367902, ranks A367906.
Set systems not of this type are counted by A367903, ranks A367907.
Set systems uniquely of this type are counted by A367904, ranks A367908.
Unlabeled multiset partitions of this type are A368098, complement A368097.
A version for MM-numbers of multisets is A368100, complement A355529.
Factorizations of this type are A368414/A370814, complement A368413/A370813.
For prime instead of binary indices we have A370586, differences of A370582.
The complement for prime indices is A370587, differences of A370583.
The complement is counted by A370589, differences of A370637.
Partial sums are A370636.
The complement has partial sums A370637/A370643, minima A370642/A370644.
The case of a unique choice is A370641, differences of A370638.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A058891 counts set-systems, A003465 covering, A323818 connected.
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, A326702, A355739, A355740, A367770, A367772, A367905, A367909, A367912, A368094, A368095, A368109, A370640.

Programs

  • Mathematica
    bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Length[Select[Subsets[Range[n]],MemberQ[#,n] && Select[Tuples[bpe/@#],UnsameQ@@#&]!={}&]],{n,0,10}]

Formula

First differences of A370636.

Extensions

a(19)-a(42) from Alois P. Heinz, Mar 09 2024

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

Views

Author

Gus Wiseman, Mar 17 2024

Keywords

Comments

The Heinz numbers of these partitions are given by A371177.
Also partitions such that the number of distinct parts is equal to the number of distinct divisors of parts.

Examples

			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)

Crossrefs

The LHS is represented by A001221, distinct case of A001222.
For partitions with no divisors of parts we have A305148, ranks A316476.
The RHS is represented by A370820, for prime factors A303975.
The strict case is A371128.
Counting all parts on the LHS gives A371130, ranks A370802.
The complement is counted by A371132.
For submultisets instead of distinct parts we have A371172, ranks A371165.
These partitions have ranks A371177.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length.
Cf. A000837, A003963, A239312, A285573, A305148, A319055, A355529, A370803, A370808, A370813, A371168, A371171, A371173.

Programs

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

A370589 Number of subsets of {1..n} containing n such that it is not possible to choose a different binary index of each element.

Original entry on oeis.org

0, 0, 0, 1, 1, 6, 17, 42, 67, 175, 400, 870, 1841, 3820, 7837, 15920, 30997, 63370, 128348, 258699, 520042, 1043284, 2090732, 4186382, 8379022, 16765549, 33540664, 67092258, 134198633, 268412631, 536844414, 1073710403, 2147296425, 4294753612, 8589686922, 17179580003
Offset: 0

Views

Author

Gus Wiseman, Mar 08 2024

Keywords

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.

Examples

			The binary indices of {1,4,5} are {{1},{3},{1,3}}, from which it is not possible to choose three different elements, so S is counted under a(3).
The binary indices of S = {1,6,8,9} are {{1},{2,3},{4},{1,4}}, from which it is not possible to choose four different elements, so S is counted under a(9).
The a(0) = 0 through a(6) = 17 subsets:
  .  .  .  {1,2,3}  {1,2,3,4}  {1,4,5}      {2,4,6}
                               {1,2,3,5}    {1,2,3,6}
                               {1,2,4,5}    {1,2,4,6}
                               {1,3,4,5}    {1,2,5,6}
                               {2,3,4,5}    {1,3,4,6}
                               {1,2,3,4,5}  {1,3,5,6}
                                            {1,4,5,6}
                                            {2,3,4,6}
                                            {2,3,5,6}
                                            {2,4,5,6}
                                            {3,4,5,6}
                                            {1,2,3,4,6}
                                            {1,2,3,5,6}
                                            {1,2,4,5,6}
                                            {1,3,4,5,6}
                                            {2,3,4,5,6}
                                            {1,2,3,4,5,6}

Links

Crossrefs

Simple graphs not of this type are counted by A133686, covering A367869.
Unlabeled graphs of this type are counted by A140637, complement A134964.
Simple graphs of this type are counted by A367867, covering A367868.
Set systems not of this type are counted by A367902, ranks A367906.
Set systems of this type are counted by A367903, ranks A367907.
Set systems uniquely not of this type are counted by A367904, ranks A367908.
Unlabeled multiset partitions of this type are A368097, complement A368098.
A version for MM-numbers of multisets is A355529, complement A368100.
Factorizations are counted by A368413/A370813, complement A368414/A370814.
The complement for prime indices is A370586, differences of A370582.
For prime indices we have A370587, differences of A370583.
Partial sums are A370637/A370643, minima A370642/A370644.
The complement is counted by A370639, partial sums A370636.
The version for a unique choice is A370641, partial sums A370638.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A058891 counts set-systems, A003465 covering, A323818 connected.
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, A326702, A355739, A355740, A367772, A367905, A367909, A367912, A368094, A368095, A368109, A370640.

Programs

  • Mathematica
    bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Length[Select[Subsets[Range[n]],MemberQ[#,n] && Select[Tuples[bpe/@#],UnsameQ@@#&]=={}&]],{n,0,10}]

Extensions

a(19)-a(35) from Alois P. Heinz, Mar 09 2024

A370804 Number of non-condensed integer partitions of n into parts > 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 3, 3, 6, 6, 12, 14, 21, 25, 37, 43, 62, 75, 101, 124, 167, 198, 261, 316, 401, 488, 618, 745, 930, 1119, 1379, 1664, 2032, 2433, 2960, 3537, 4259, 5076, 6094, 7227, 8629, 10205, 12126, 14302, 16932, 19893, 23471, 27502, 32315, 37775
Offset: 0

Views

Author

Gus Wiseman, Mar 03 2024

Keywords

Comments

These are partitions without ones such that it is not possible to choose a different divisor of each part.

Examples

			The a(6) = 1 through a(14) = 12 partitions:
  (222)  .  (2222)  (333)   (3322)   (3332)   (3333)    (4333)    (4442)
                    (3222)  (4222)   (5222)   (4422)    (7222)    (5333)
                            (22222)  (32222)  (6222)    (33322)   (5522)
                                              (33222)   (43222)   (8222)
                                              (42222)   (52222)   (33332)
                                              (222222)  (322222)  (43322)
                                                                  (44222)
                                                                  (53222)
                                                                  (62222)
                                                                  (332222)
                                                                  (422222)
                                                                  (2222222)

Crossrefs

These partitions have as ranks the odd terms of A355740.
The version with ones is A370320, complement A239312.
The complement without ones is A370805.
The version for prime factors is A370807, with ones A370593.
The version for factorizations is A370813, complement A370814.
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.
Cf. A355529, A355739, A367867, A367901, A368110, A368413, A370595, A370806, A370808, A370810.

Programs

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

Extensions

More terms from Jinyuan Wang, Feb 14 2025
Previous Showing 11-20 of 31 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.