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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A370640 Number of maximal subsets of {1..n} such that it is possible to choose a different binary index of each element.

Original entry on oeis.org

1, 1, 1, 3, 3, 8, 17, 32, 32, 77, 144, 242, 383, 580, 843, 1201, 1201, 2694, 4614, 7096, 10219, 14186, 19070, 25207, 32791, 42160, 53329, 66993, 82811, 101963, 124381, 151286, 151286, 324695, 526866, 764438, 1038089, 1358129, 1725921, 2154668, 2640365, 3202985
Offset: 0

Views

Author

Gus Wiseman, Mar 10 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.
Also choices of A070939(n) elements of {1..n} such that it is possible to choose a different binary index of each.

Examples

			The a(0) = 1 through a(6) = 17 subsets:
  {}  {1}  {1,2}  {1,2}  {1,2,4}  {1,2,4}  {1,2,4}
                  {1,3}  {1,3,4}  {1,2,5}  {1,2,5}
                  {2,3}  {2,3,4}  {1,3,4}  {1,2,6}
                                  {1,3,5}  {1,3,4}
                                  {2,3,4}  {1,3,5}
                                  {2,3,5}  {1,3,6}
                                  {2,4,5}  {1,4,6}
                                  {3,4,5}  {1,5,6}
                                           {2,3,4}
                                           {2,3,5}
                                           {2,3,6}
                                           {2,4,5}
                                           {2,5,6}
                                           {3,4,5}
                                           {3,4,6}
                                           {3,5,6}
                                           {4,5,6}
The a(0) = 1 through a(6) = 17 set-systems:
    {1}  {1}{2}  {1}{2}   {1}{2}{3}   {1}{2}{3}    {1}{2}{3}
                 {1}{12}  {1}{12}{3}  {1}{12}{3}   {1}{12}{3}
                 {2}{12}  {2}{12}{3}  {1}{2}{13}   {1}{2}{13}
                                      {2}{12}{3}   {1}{2}{23}
                                      {2}{3}{13}   {1}{3}{23}
                                      {1}{12}{13}  {2}{12}{3}
                                      {12}{3}{13}  {2}{3}{13}
                                      {2}{12}{13}  {1}{12}{13}
                                                   {1}{12}{23}
                                                   {1}{13}{23}
                                                   {12}{3}{13}
                                                   {12}{3}{23}
                                                   {2}{12}{13}
                                                   {2}{12}{23}
                                                   {2}{13}{23}
                                                   {3}{13}{23}
                                                   {12}{13}{23}

Crossrefs

Dominated by A357812.
The version for set-systems is A368601, max of A367902 (complement A367903).
For prime indices we have A370585, with n A370590, see also A370591.
This is the maximal case of A370636 (complement A370637).
The case of a unique choice is A370638.
The case containing n is A370641, non-maximal A370639.
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.
A307984 counts Q-bases of logarithms of positive integers.
A355741 counts choices of a prime factor of each prime index.
Cf. A133686, A326031, A326702, A367905, A367909, A367912, A368109, A368110, A370592, A370642.

Programs

  • Mathematica
    bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Length[Select[Subsets[Range[n],{IntegerLength[n,2]}], Select[Tuples[bpe/@#],UnsameQ@@#&]!={}&]],{n,0,10}]
  • PARI
    lista(nn) = my(b, m=Map(Mat([[[]], 1])), t, u, v, w, z); for(n=0, nn, t=Mat(m)~; b=Vecrev(binary(n)); u=select(i->b[i], [1..#b]); for(i=1, #t, v=t[1, i]; w=List([]); for(j=1, #v, for(k=1, #u, if(!setsearch(v[j], u[k]), listput(w, setunion(v[j], [u[k]]))))); w=Set(w); if(#w, z=0; mapisdefined(m, w, &z); mapput(m, w, z+t[2, i]))); print1(mapget(m, [[1..#b]]), ", ")); \\ Jinyuan Wang, Mar 28 2025

Extensions

More terms from Jinyuan Wang, Mar 28 2025

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}]

A370645 Number of integer factorizations of n into unordered factors > 1 such that only one set can be obtained by choosing a different prime factor of each factor.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1
Offset: 1

Views

Author

Gus Wiseman, Mar 01 2024

Keywords

Comments

All of these factorizations are co-balanced (A340596).

Examples

			The factorization f = (3*6*10) has prime factor choices (3,2,2), (3,3,2), (3,2,5), and (3,3,5), of which only (3,2,5) has all different parts, so f is counted under a(180).
The a(n) factorizations for n = 2, 12, 24, 36, 72, 120, 144, 180, 288:
  (2)  (2*6)  (3*8)   (4*9)   (8*9)   (3*5*8)   (2*72)   (4*5*9)   (3*96)
       (3*4)  (4*6)   (6*6)   (2*36)  (4*5*6)   (3*48)   (5*6*6)   (4*72)
              (2*12)  (2*18)  (3*24)  (2*3*20)  (4*36)   (2*3*30)  (6*48)
                      (3*12)  (4*18)  (2*5*12)  (6*24)   (2*5*18)  (8*36)
                              (6*12)  (2*6*10)  (8*18)   (2*6*15)  (9*32)
                                      (3*4*10)  (9*16)   (2*9*10)  (12*24)
                                                (12*12)  (3*4*15)  (16*18)
                                                         (3*5*12)  (2*144)
                                                         (3*6*10)

Crossrefs

Multisets of this type are ranked by A368101, see also A368100, A355529.
For nonexistence we have A368413, complement A368414.
Subsets of this type are counted by A370584, see also A370582, A370583.
Maximal sets of this type are counted by A370585.
The version for partitions is A370594, see also A370592, A370593.
Subsets of this type are counted by A370638, see also A370636, A370637.
For unlabeled multiset partitions we have A370646, also A368098, A368097.
A001055 counts factorizations, strict A045778.
A006530 gives greatest prime factor, least A020639.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A027746 lists prime factors, A112798 indices, length A001222.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
A355741 counts ways to choose a prime factor of each prime index.
For set-systems see A367902-A367908.
Cf. A000040, A000720, A003963, A340596, A340653, A355744, A355745, A368110.

Programs

  • 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[Union[Sort/@Select[Tuples[First /@ FactorInteger[#]&/@#], UnsameQ@@#&]]]==1&]],{n,100}]

A370647 Numbers such that only one set can be obtained by choosing a different prime factor of each prime index.

Original entry on oeis.org

1, 3, 5, 7, 11, 15, 17, 19, 23, 31, 33, 35, 39, 41, 51, 53, 55, 59, 65, 67, 69, 77, 83, 85, 87, 91, 93, 95, 97, 103, 109, 111, 119, 123, 127, 129, 131, 155, 157, 161, 165, 169, 177, 179, 183, 185, 187, 191, 201, 203, 205, 209, 211, 213, 217, 227, 235, 237, 241
Offset: 1

Views

Author

Gus Wiseman, Mar 06 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 91 are {4,6}, with only choice {2,3}, so 91 is in the sequence.
The terms together with their prime indices begin:
     1: {}        53: {16}      109: {29}
     3: {2}       55: {3,5}     111: {2,12}
     5: {3}       59: {17}      119: {4,7}
     7: {4}       65: {3,6}     123: {2,13}
    11: {5}       67: {19}      127: {31}
    15: {2,3}     69: {2,9}     129: {2,14}
    17: {7}       77: {4,5}     131: {32}
    19: {8}       83: {23}      155: {3,11}
    23: {9}       85: {3,7}     157: {37}
    31: {11}      87: {2,10}    161: {4,9}
    33: {2,5}     91: {4,6}     165: {2,3,5}
    35: {3,4}     93: {2,11}    169: {6,6}
    39: {2,6}     95: {3,8}     177: {2,17}
    41: {13}      97: {25}      179: {41}
    51: {2,7}    103: {27}      183: {2,18}

Crossrefs

For nonexistence we have A355529, count A370593.
For binary instead of prime indices we have A367908, counted by A367904.
For existence we have A368100, count A370592.
For a sequence instead of set of factors we have A368101.
The version for subsets is A370584, see also A370582, A370583.
Maximal sets of this type are counted by A370585.
Partitions of this type are counted by A370594.
For subsets and binary indices we have A370638.
The version for factorizations is A370645, see also A368414, A368413.
For divisors instead of factors we have A370810, counted by A370595.
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, A355739, A355740, A355744, A355745, A368110.

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[prix/@prix[#]],UnsameQ@@#&]]]==1&]

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

Views

Author

Gus Wiseman, Mar 16 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 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}

Crossrefs

The complement is A370348, counted by A371171.
The case of equality is A370802, counted by A371130, strict A371128.
The RHS is A370820, for prime factors instead of divisors A303975.
The strict version is A371168 counted by A371173.
The opposite version is A371169.
Choosable partitions: A239312 (A368110), A355740 (A370320), A370592 (A368100), A370593 (A355529).
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, A319899, A355737, A355739, A355741, A370808, A371127.

Programs

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

A370591 Number of minimal subsets of {1..n} such that it is not possible to choose a different prime factor of each element (non-choosable).

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 4, 4, 7, 11, 16, 16, 30, 30, 39, 73
Offset: 0

Views

Author

Gus Wiseman, Feb 28 2024

Keywords

Examples

			The a(1) = 1 through a(10) = 16 subsets:
{1}  {1}  {1}  {1}    {1}    {1}      {1}      {1}      {1}      {1}
               {2,4}  {2,4}  {2,4}    {2,4}    {2,4}    {2,4}    {2,4}
                             {2,3,6}  {2,3,6}  {2,8}    {2,8}    {2,8}
                             {3,4,6}  {3,4,6}  {4,8}    {3,9}    {3,9}
                                               {2,3,6}  {4,8}    {4,8}
                                               {3,4,6}  {2,3,6}  {2,3,6}
                                               {3,6,8}  {2,6,9}  {2,6,9}
                                                        {3,4,6}  {3,4,6}
                                                        {3,6,8}  {3,6,8}
                                                        {4,6,9}  {4,6,9}
                                                        {6,8,9}  {6,8,9}
                                                                 {2,5,10}
                                                                 {4,5,10}
                                                                 {5,8,10}
                                                                 {3,5,6,10}
                                                                 {5,6,9,10}

Crossrefs

Minimal case of A370583, complement A370582.
For binary indices instead of factors we have A370642, minima of A370637.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, indices A112798, length A001222.
A355741 counts choices of a prime factor of each prime index.
A367902 counts choosable set-systems, ranks A367906, unlabeled A368095.
A367903 counts non-choosable set-systems, ranks A367907, unlabeled A368094.
A368098 counts choosable unlabeled multiset partitions, complement A368097.
A368100 ranks choosable multisets, complement A355529.
A368414 counts choosable factorizations, complement A368413.
A370585 counts maximal choosable sets.
A370592 counts choosable partitions, complement A370593.
Cf. A000040, A000720, A045778, A133686, A355739, A355744, A355745, A367771, A370584, A370586, A370587, A370589.

Programs

  • Mathematica
    Table[Length[fasmin[Select[Subsets[Range[n]], Length[Select[Tuples[prix/@#],UnsameQ@@#&]]==0&]]], {n,0,15}]

A370805 Number of condensed integer partitions of n into parts > 1.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 4, 6, 6, 9, 11, 15, 18, 22, 27, 34, 41, 51, 62, 75, 90, 109, 129, 153, 185, 217, 258, 307, 359, 421, 493, 577, 675, 788, 909, 1062, 1227, 1418, 1633, 1894, 2169, 2497, 2860, 3285, 3754, 4298, 4894, 5587, 6359, 7230, 8215, 9331, 10567, 11965
Offset: 0

Views

Author

Gus Wiseman, Mar 04 2024

Keywords

Comments

These are partitions without ones such that it is possible to choose a different divisor of each part.

Examples

			The a(0) = 1 through a(9) = 6 partitions:
  ()  .  (2)  (3)  (4)    (5)    (6)    (7)      (8)      (9)
                   (2,2)  (3,2)  (3,3)  (4,3)    (4,4)    (5,4)
                                 (4,2)  (5,2)    (5,3)    (6,3)
                                        (3,2,2)  (6,2)    (7,2)
                                                 (3,3,2)  (4,3,2)
                                                 (4,2,2)  (5,2,2)

Crossrefs

The version with ones is A239312, complement A370320.
These partitions have as ranks the odd terms of A368110, complement A355740.
The version for prime factors is A370592, complement A370593, post A370807.
The complement without ones is A370804, ranked by the odd terms of A355740.
The version for factorizations is A370814, complement A370813.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A355731 counts choices of a divisor of each prime index, firsts A355732.
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

A370809 Greatest number of multisets that can be obtained by choosing a prime factor of each part of an integer partition of n.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 3, 4, 3, 4, 4, 4, 4, 6, 4, 6, 6, 6, 6, 8, 6, 8, 8, 9, 8, 10, 9, 12, 10, 12, 12, 12, 12, 16, 13, 16, 16, 18, 16, 20, 18, 20, 20, 24, 20, 24, 24, 24, 26, 30, 26, 30, 30, 32, 32, 36, 32, 36, 36, 40, 38, 42, 40, 45, 44, 48
Offset: 0

Views

Author

Gus Wiseman, Mar 05 2024

Keywords

Examples

			For the partition (10,6,3,2) there are 4 choices: {2,2,2,3}, {2,2,3,3}, {2,2,3,5}, {2,3,3,5} so a(21) >= 4.
For the partitions of 6 we have the following choices:
  (6): {{2},{3}}
  (51): {}
  (42): {{2,2}}
  (411): {}
  (33): {{3,3}}
  (321): {}
  (3111): {}
  (222): {{2,2,2}}
  (2211): {}
  (21111): {}
  (111111): {}
So a(6) = 2.

Crossrefs

For just all divisors (not just prime factors) we have A370808.
The version for factorizations is A370817, for all divisors A370816.
A000041 counts integer partitions, strict A000009.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A355741, A355744, A355745 choose prime factors of prime indices.
A368413 counts non-choosable factorizations, complement A368414.
A370320 counts non-condensed partitions, ranks A355740.
A370592, A370593, A370594, `A370807 count non-choosable partitions.
Cf. A000792, A048249, A063834, A239312, A319055, A339095, A355529, A355733, A367771, A368100, A370585.

Programs

  • Mathematica
    Table[Max[Length[Union[Sort /@ Tuples[If[#==1,{},First/@FactorInteger[#]]& /@ #]]]&/@IntegerPartitions[n]],{n,0,30}]

Extensions

Terms a(31) onward from Max Alekseyev, Sep 17 2024

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

Views

Author

Gus Wiseman, Mar 16 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:
     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}

Crossrefs

The strict version is A370348 counted by A371171.
The case of equality is A370802, counted by A371130, strict A371128.
The RHS is A370820, for prime factors instead of divisors A303975.
The complement is A371168, counted by A371173.
The opposite version is A371170.
The version for prime factors instead of divisors on the RHS is A319899.
Choosable partitions: A239312 (A368110), A355740 (A370320), A370592 (A368100), A370593 (A355529).
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, A355737, A355739, A355741, A370808, A370813, A370814, A371127.

Programs

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

A387118 Number of integer partitions of n without choosable initial intervals.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 6, 8, 13, 19, 28, 37, 52, 70, 97, 130, 172, 224, 293, 378, 492, 630, 806, 1018, 1286, 1609, 2019, 2514, 3131, 3874, 4784, 5872, 7198, 8786, 10712, 13013, 15794, 19100, 23063, 27752, 33341, 39939, 47781, 57013, 67955, 80816, 95992, 113773, 134668
Offset: 0

Views

Author

Gus Wiseman, Sep 02 2025

Keywords

Comments

The initial interval of a nonnegative integer x is the set {1,...,x}.
We say that a sequence of nonempty sets is choosable iff it is possible to choose a different element from each set. For example, ({1,2},{1},{1,3}) is choosable because we have the choice (2,1,3), but ({1},{2},{1,3},{2,3}) is not.

Examples

			The partition y = (2,2,1) has initial intervals ({1,2},{1,2},{1}), which are not choosable, so y is counted under a(5).
The a(2) = 1 through a(8) = 13 partitions:
  (11)  (111)  (211)   (221)    (222)     (511)      (611)
               (1111)  (311)    (411)     (2221)     (2222)
                       (2111)   (2211)    (3211)     (3221)
                       (11111)  (3111)    (4111)     (3311)
                                (21111)   (22111)    (4211)
                                (111111)  (31111)    (5111)
                                          (211111)   (22211)
                                          (1111111)  (32111)
                                                     (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

Crossrefs

The complement is counted by A238873, ranks A387112.
The complement for divisors is A239312, ranks A368110.
For divisors instead of initial intervals we have A370320, ranks A355740.
The complement for prime factors is A370592, ranks A368100.
For prime factors instead of initial intervals we have A370593, ranks A355529.
These partitions have ranks A387113.
For partitions instead of initial intervals we have A387134.
The complement for partitions is A387328.
For strict partitions instead of initial intervals we have A387137, ranks A387176.
The complement for strict partitions is A387178.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A367902 counts choosable set-systems, complement A367903.
A370582 counts sets with choosable prime factors, complement A370583.
Cf. A005703, A052335, A367867, A367901, A367905, A367907, A370585, A370594, A387111.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Select[Tuples[Range/@#],UnsameQ@@#&]=={}&]],{n,0,10}]

Extensions

More terms from Jinyuan Wang, Sep 05 2025
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