A355744 -id:A355744 - cp's OEIS frontend

A370803 Number of integer partitions of n such that more than one set can be obtained by choosing a different divisor of each part.

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

Gus Wiseman, Mar 03 2024

nonn

			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

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.

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

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

More terms from Jinyuan Wang, Feb 14 2025

A355735 Number of ways to choose a divisor of each prime index of n (taken in weakly increasing order) such that the result is weakly increasing.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 3, 2, 2, 2, 4, 3, 3, 1, 2, 3, 4, 2, 5, 2, 3, 2, 3, 4, 4, 3, 4, 3, 2, 1, 3, 2, 4, 3, 6, 4, 7, 2, 2, 5, 4, 2, 4, 3, 4, 2, 6, 3, 3, 4, 5, 4, 3, 3, 7, 4, 2, 3, 6, 2, 7, 1, 6, 3, 2, 2, 5, 4, 6, 3, 4, 6, 4, 4, 4, 7, 4, 2, 5, 2, 2, 5, 3, 4, 7

Offset: 1

Gus Wiseman, Jul 16 2022

nonn

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.

			The a(15) = 3 ways are: (1,1), (1,3), (2,3).
The a(18) = 3 ways are: (1,1,1), (1,1,2), (1,2,2).
The a(2) = 1 through a(19) = 4 ways:
  1  1  11  1  11  1  111  11  11  1  111  1  11  11  1111  1  111  1
     2      3  12  2       12  13  5  112  2  12  13        7  112  2
                   4       22              3  14  23           122  4
                                           6                        8

Wikipedia, Cartesian product.

Allowing any choice of divisors gives A355731, firsts A355732.
Choosing a multiset instead of sequence gives A355733, firsts A355734.
Positions of first appearances are A355736.
Choosing only prime divisors gives A355745, variations A355741, A355744.
The reverse version is A355749.
A000005 counts divisors.
A001414 adds up distinct prime divisors, counted by A001221.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A061395 selects the maximum prime index.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
Cf. A000720, A076610, A316524, A335433, A335448, A340827, A340852, A344616, A355737, A355739, A355740, A355742.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Tuples[Divisors/@primeMS[n]],LessEqual@@#&]],{n,100}]

A370586 Number of subsets of {1..n} containing n such that it is possible to choose a different prime factor of each element (choosable).

Original entry on oeis.org

0, 0, 1, 2, 2, 6, 8, 20, 12, 20, 44, 116, 88, 320, 380, 508, 264, 1792, 968, 4552, 3136, 5600, 10056, 27896, 11792, 16384, 46688, 19584, 48288, 198528, 110928, 507984, 99648, 463552, 859376, 821136, 470688, 3730368, 4033920, 4651296, 2932512, 19078464

Offset: 0

Gus Wiseman, Feb 26 2024

nonn

			The a(0) = 0 through a(7) = 20 subsets:
  .  .  {2}  {3}    {4}    {5}      {6}      {7}
             {2,3}  {3,4}  {2,5}    {2,6}    {2,7}
                           {3,5}    {3,6}    {3,7}
                           {4,5}    {4,6}    {4,7}
                           {2,3,5}  {5,6}    {5,7}
                           {3,4,5}  {2,5,6}  {6,7}
                                    {3,5,6}  {2,3,7}
                                    {4,5,6}  {2,5,7}
                                             {2,6,7}
                                             {3,4,7}
                                             {3,5,7}
                                             {3,6,7}
                                             {4,5,7}
                                             {4,6,7}
                                             {5,6,7}
                                             {2,3,5,7}
                                             {2,5,6,7}
                                             {3,4,5,7}
                                             {3,5,6,7}
                                             {4,5,6,7}

First differences of A370582, complement A370583, cf. A370584.
Maximal choosable sets are counted by A370585.
The complement is counted by A370587.
For a unique choice we have A370588.
For binary indices instead of prime factors we have A370639.
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.
A370592 counts choosable partitions, complement A370593.
Cf. A000040, A000720, A005117, A045778, A133686, A355739, A355744, A355745, A367771, A367905, A370636.

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

a(19)-a(41) from Alois P. Heinz, Feb 27 2024

A355734 Least k such that there are exactly n multisets that can be obtained by choosing a divisor of each prime index of k.

Original entry on oeis.org

1, 3, 7, 13, 21, 35, 39, 89, 133, 105, 91, 195, 351, 285, 247, 333, 273, 481, 455, 555, 623, 801, 791, 741, 1359, 1157, 1281, 1335, 1365, 1443, 1977, 1729, 1967, 1869, 2109, 3185, 2373, 2769, 2639, 4361, 3367, 3653, 3885, 3471, 4613, 5883, 5187, 5551, 6327

Offset: 1

Gus Wiseman, Jul 21 2022

nonn

This is the position of first appearance of n in A355733.

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.

			The terms together with their prime indices begin:
    1: {}
    3: {2}
    7: {4}
   13: {6}
   21: {2,4}
   35: {3,4}
   39: {2,6}
   89: {24}
  133: {4,8}
  105: {2,3,4}
   91: {4,6}
  195: {2,3,6}
  351: {2,2,2,6}
For example, the choices for a(12) = 195 are:
  {1,1,1}  {1,2,2}  {1,3,6}
  {1,1,2}  {1,2,3}  {2,2,3}
  {1,1,3}  {1,2,6}  {2,3,3}
  {1,1,6}  {1,3,3}  {2,3,6}

Counting all choices of divisors gives A355732, firsts of A355731.
Positions of first appearances in A355733.
Choosing weakly increasing divisors gives A355736, firsts of A355735.
A000005 counts divisors.
A001414 adds up distinct prime divisors, counted by A001221.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
Cf. A000720, A076610, A340852, A344606, A355737, A355739, A355740, A355741, A355744, A355747.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
az=Table[Length[Union[Sort/@Tuples[Divisors/@primeMS[n]]]],{n,1000}];
Table[Position[az,k][[1,1]],{k,mnrm[az]}]

A355535 Odd numbers of which it is not possible to choose a different prime factor of each prime index.

Original entry on oeis.org

9, 21, 25, 27, 45, 49, 57, 63, 75, 81, 99, 105, 115, 117, 121, 125, 133, 135, 147, 153, 159, 171, 175, 189, 195, 207, 225, 231, 243, 245, 261, 273, 275, 279, 285, 289, 297, 315, 325, 333, 343, 345, 351, 357, 361, 363, 369, 371, 375, 387, 393, 399, 405, 423

Offset: 1

Gus Wiseman, Jul 22 2022

nonn

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.

			The terms together with their prime indices begin:
    9: {2,2}
   21: {2,4}
   25: {3,3}
   27: {2,2,2}
   45: {2,2,3}
   49: {4,4}
   57: {2,8}
   63: {2,2,4}
   75: {2,3,3}
   81: {2,2,2,2}
   99: {2,2,5}
  105: {2,3,4}
For example, the prime indices of 897 are {2,6,9}, of which we can choose prime factors in two ways: (2,2,3) or (2,3,3); but neither of these has all distinct elements, so 897 is in the sequence.

Including evens gives A355529.
The version for all divisors including evens is A355740, zeros of A355739.
Choices of a prime factor of each prime index: A355741, unordered A355744.
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
A120383 lists numbers divisible by all of their prime indices.
Cf. A000720, A076610, A289509, A302796, A327486, A355731, A355733, A355742.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],OddQ[#]&&Select[Tuples[primeMS/@primeMS[#]],UnsameQ@@#&]=={}&]

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

Gus Wiseman, Mar 03 2024

nonn

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

			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

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.

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

More terms from Jinyuan Wang, Feb 14 2025

A355747 Number of multisets that can be obtained by choosing a divisor of each positive integer from 1 to n.

Original entry on oeis.org

1, 1, 2, 4, 10, 20, 58, 116, 320, 772, 2170, 4340, 14112, 28224, 78120, 212004, 612232, 1224464, 3873760, 7747520, 24224608, 64595088, 175452168, 350904336

Offset: 0

Gus Wiseman, Jul 20 2022

			The a(0) = 1 through a(4) = 10 multisets:
  {}  {1}  {1,1}  {1,1,1}  {1,1,1,1}
           {1,2}  {1,1,2}  {1,1,1,2}
                  {1,1,3}  {1,1,1,3}
                  {1,2,3}  {1,1,1,4}
                           {1,1,2,2}
                           {1,1,2,3}
                           {1,1,2,4}
                           {1,1,3,4}
                           {1,2,2,3}
                           {1,2,3,4}

The sum of the same integers is A000096.
The product of the same integers is A000142, Heinz number A070826.
Counting sequences instead of multisets gives A066843.
The integers themselves are the rows of A131818 (shifted).
For prime indices we have A355733, only prime factors A355744.
For prime factors instead of divisors we have A355746, factors A355537.
A000005 counts divisors.
A000040 lists the prime numbers.
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
Cf. A000720, A002110, A006218, A076610, A327486, A355538, A355731, A355737, A355741, A355745.

Table[Length[Union[Sort/@Tuples[Divisors/@Range[n]]]],{n,0,10}]

from sympy import divisors
from itertools import count, islice
def agen():
    s = {tuple()}
    for n in count(1):
        yield len(s)
        s = set(tuple(sorted(t+(d,))) for t in s for d in divisors(n))
print(list(islice(agen(), 16))) # Michael S. Branicky, Aug 03 2022

a(n) = A355733(A070826(n)).

a(p) = 2*a(p-1) for p prime. - Michael S. Branicky, Aug 03 2022

a(15)-a(21) from Michael S. Branicky, Aug 03 2022

a(22)-a(23) from Michael S. Branicky, Aug 08 2022

A370587 Number of subsets of {1..n} containing 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, 2, 6, 10, 24, 44, 116, 236, 468, 908, 1960, 3776, 7812, 15876, 32504, 63744, 130104, 257592, 521152, 1042976, 2087096, 4166408, 8376816, 16760832, 33507744, 67089280, 134169440, 268236928, 536759984, 1073233840, 2147384000, 4294503744, 8589075216, 17179048048

Offset: 0

Gus Wiseman, Feb 28 2024

nonn

			The a(0) = 0 through a(5) = 10 subsets:
  .  {1}  {1,2}  {1,3}    {1,4}      {1,5}
                 {1,2,3}  {2,4}      {1,2,5}
                          {1,2,4}    {1,3,5}
                          {1,3,4}    {1,4,5}
                          {2,3,4}    {2,4,5}
                          {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}

First differences of A370583, complement A370582, cf. A370584.
The complement is counted by A370586.
For a unique choice we have A370588.
For binary indices instead of factors we have A370639, complement A370589.
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, A005117, A045778, A133686, A355739, A355744, A355745, A367905, A370636.

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

More terms from Jinyuan Wang, Mar 28 2025

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

Gus Wiseman, Mar 05 2024

nonn

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.

			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}

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.

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&]

A356065 Squarefree numbers whose prime indices are all prime-powers.

Original entry on oeis.org

1, 3, 5, 7, 11, 15, 17, 19, 21, 23, 31, 33, 35, 41, 51, 53, 55, 57, 59, 67, 69, 77, 83, 85, 93, 95, 97, 103, 105, 109, 115, 119, 123, 127, 131, 133, 155, 157, 159, 161, 165, 177, 179, 187, 191, 201, 205, 209, 211, 217, 227, 231, 241, 249, 253, 255, 265, 277

Offset: 1

Gus Wiseman, Jul 25 2022

nonn

			105 has prime indices {2,3,4}, all three of which are prime-powers, so 105 is in the sequence.

The multiplicative version (factorizations) is A050361, non-strict A000688.
Heinz numbers of the partitions counted by A054685, with 1's A106244, non-strict A023894, non-strict with 1's A023893.
Counting twice-partitions of this type gives A279786, non-strict A279784.
Counting twice-factorizations gives A295935, non-strict A296131.
These are the odd products of distinct elements of A302493.
Allowing prime index 1 gives A302496, non-strict A302492.
The case of primes (instead of prime-powers) is A302590, non-strict A076610.
These are the squarefree positions of 1's in A355741.
This is the squarefree case of A355743, complement A356066.
A001222 counts prime-power divisors.
A005117 lists the squarefree numbers.
A034699 gives maximal prime-power divisor.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
A355742 chooses a prime-power divisor of each prime index.
Cf. A001970, A055887, A063834, A302601, A355731, A355744, A356064.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],SquareFreeQ[#]&&And@@PrimePowerQ/@primeMS[#]&]

Intersection of A005117 and A355743.

cp's OEIS Frontend

A370803 Number of integer partitions of n such that more than one set can be obtained by choosing a different divisor of each part.

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A355735 Number of ways to choose a divisor of each prime index of n (taken in weakly increasing order) such that the result is weakly increasing.

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A370586 Number of subsets of {1..n} containing n such that it is possible to choose a different prime factor of each element (choosable).

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A355734 Least k such that there are exactly n multisets that can be obtained by choosing a divisor of each prime index of k.

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A355535 Odd numbers of which it is not possible to choose a different prime factor of each prime index.

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A370595 Number of integer partitions of n such that only one set can be obtained by choosing a different divisor of each part.

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A355747 Number of multisets that can be obtained by choosing a divisor of each positive integer from 1 to n.

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A370587 Number of subsets of {1..n} containing n such that it is not possible to choose a different prime factor of each element (non-choosable).

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A370810 Numbers n such that only one set can be obtained by choosing a different divisor of each prime index of n.

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