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.

Showing 1-10 of 49 results. Next

A368100 Numbers of which it is possible to choose a different prime factor of each prime index.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 39, 41, 43, 47, 51, 53, 55, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 107, 109, 111, 113, 119, 123, 127, 129, 131, 137, 139, 141, 143, 145, 149, 151, 155, 157, 161, 163
Offset: 1

Views

Author

Gus Wiseman, Dec 12 2023

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 2849 are {4,5,12}, with prime factors {{2,2},{5},{2,2,3}}, and of the two choices (2,5,2) and (2,5,3) the latter has all different terms, so 2849 is in the sequence.
The terms together with their prime indices of prime indices begin:
   1: {}
   3: {{1}}
   5: {{2}}
   7: {{1,1}}
  11: {{3}}
  13: {{1,2}}
  15: {{1},{2}}
  17: {{4}}
  19: {{1,1,1}}
  23: {{2,2}}
  29: {{1,3}}
  31: {{5}}
  33: {{1},{3}}
  35: {{2},{1,1}}
  37: {{1,1,2}}
  39: {{1},{1,2}}

Crossrefs

The complement is A355529, odd A355535, binary A367907.
Positions of positive terms in A367771.
The version for binary indices is A367906, positive positions in A367905.
For a unique choice we have A368101, binary A367908.
The version for divisors instead of factors is A368110, complement A355740.
A058891 counts set-systems, covering A003465, connected A323818.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
Cf. A092918, A355737, A355739, A355741, A355744, A355745, A367902.

Programs

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

A239312 Number of condensed integer partitions of n.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 5, 6, 9, 10, 14, 16, 23, 27, 33, 41, 51, 62, 75, 93, 111, 134, 159, 189, 226, 271, 317, 376, 445, 520, 609, 714, 832, 972, 1129, 1304, 1520, 1753, 2023, 2326, 2692, 3077, 3540, 4050, 4642, 5298, 6054, 6887, 7854, 8926, 10133, 11501, 13044
Offset: 0

Views

Author

Clark Kimberling, Mar 15 2014

Keywords

Comments

Suppose that p is a partition of n. Let x(1), x(2), ..., x(k) be the distinct parts of p, and let m(i) be the multiplicity of x(i) in p. Let c(p) be the partition {m(1)*x(1), m(2)*x(2), ..., x(k)*m(k)} of n. Call a partition q of n a condensed partition of n if q = c(p) for some partition p of n. Then a(n) is the number of distinct condensed partitions of n. Note that c(p) = p if and only if p has distinct parts and that condensed partitions can have repeated parts.
Also the number of integer partitions of n such that it is possible to choose a different divisor of each part. For example, the partition (6,4,4,1) has choices (3,2,4,1), (3,4,2,1), (6,2,4,1), (6,4,2,1) so is counted under a(15). - Gus Wiseman, Mar 12 2024

Examples

			a(5) = 3 gives the number of partitions of 5 that result from condensations as shown here: 5 -> 5, 41 -> 41, 32 -> 32, 311 -> 32, 221 -> 41, 2111 -> 32, 11111 -> 5.
From _Gus Wiseman_, Mar 12 2024: (Start)
The a(1) = 1 through a(9) = 10 condensed partitions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)      (9)
            (2,1)  (2,2)  (3,2)  (3,3)    (4,3)    (4,4)    (5,4)
                   (3,1)  (4,1)  (4,2)    (5,2)    (5,3)    (6,3)
                                 (5,1)    (6,1)    (6,2)    (7,2)
                                 (3,2,1)  (3,2,2)  (7,1)    (8,1)
                                          (4,2,1)  (3,3,2)  (4,3,2)
                                                   (4,2,2)  (4,4,1)
                                                   (4,3,1)  (5,2,2)
                                                   (5,2,1)  (5,3,1)
                                                            (6,2,1)
(End)

Links

Crossrefs

The strict case is A000009.
These partitions have ranks A368110, complement A355740.
The complement is counted by A370320.
The version for prime factors (not all divisors) is A370592, ranks A368100.
The complement for prime factors is A370593, ranks A355529.
For a unique choice we have A370595, ranks A370810.
For multiple choices we have A370803, ranks A370811.
The case without ones is A370805, complement A370804.
The version for factorizations is A370814, complement A370813.
A000005 counts divisors.
A000041 counts integer partitions.
A237685 counts partitions of depth 1, or A353837 if we include depth 0.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Cf. A355535, A355733, A355739, A367867, A368097, A368414, A370583, A370584, A370594, A370806, A370808.

Programs

  • Maple
    b:= proc(n,i) option remember; `if`(n=0, {[]},
      `if`(i=1, {[n]}, {seq(map(x-> `if`(j=0, x,
       sort([x[], i*j])), b(n-i*j, i-1))[], j=0..n/i)}))
    end:
a:= n-> nops(b(n$2)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 01 2019
  • Mathematica
    u[n_, k_] := u[n, k] = Map[Total, Split[IntegerPartitions[n][[k]]]]; t[n_] := t[n] = DeleteDuplicates[Table[Sort[u[n, k]], {k, 1, PartitionsP[n]}]]; Table[Length[t[n]], {n, 0,   30}]
Table[Length[Select[IntegerPartitions[n], Length[Select[Tuples[Divisors/@#],UnsameQ@@#&]]>0&]], {n,0,30}] (* Gus Wiseman, Mar 12 2024 *)

Extensions

Typo in definition corrected by Manfred Scheucher, May 29 2015
Name edited by Gus Wiseman, Mar 13 2024

A051026 Number of primitive subsequences of {1, 2, ..., n}.

Original entry on oeis.org

1, 2, 3, 5, 7, 13, 17, 33, 45, 73, 103, 205, 253, 505, 733, 1133, 1529, 3057, 3897, 7793, 10241, 16513, 24593, 49185, 59265, 109297, 163369, 262489, 355729, 711457, 879937, 1759873, 2360641, 3908545, 5858113, 10534337, 12701537, 25403073, 38090337, 63299265, 81044097, 162088193, 205482593, 410965185, 570487233, 855676353
Offset: 0

Views

Author

Eric W. Weisstein

Keywords

Comments

a(n) counts all subsequences of {1, ..., n} in which no term divides any other. If n is a prime a(n) = 2*a(n-1)-1 because for each subsequence s counted by a(n-1) two different subsequences are counted by a(n): s and s,n. There is only one exception: 1,n is not a primitive subsequence because 1 divides n. For all n>1: a(n) < 2*a(n-1). - Alois P. Heinz, Mar 07 2011
Maximal primitive subsets are counted by A326077. - Gus Wiseman, Jun 07 2019

Examples

			a(4) = 7, the primitive subsequences (including the empty sequence) are: (), (1), (2), (3), (4), (2,3), (3,4).
a(5) = 13 = 2*7-1, the primitive subsequences are: (), (5), (1), (2), (2,5), (3), (3,5), (4), (4,5), (2,3), (2,3,5), (3,4), (3,4,5).
From _Gus Wiseman_, Jun 07 2019: (Start)
The a(0) = 1 through a(5) = 13 primitive (pairwise indivisible) subsets:
  {}  {}   {}   {}     {}     {}
      {1}  {1}  {1}    {1}    {1}
           {2}  {2}    {2}    {2}
                {3}    {3}    {3}
                {2,3}  {4}    {4}
                       {2,3}  {5}
                       {3,4}  {2,3}
                              {2,5}
                              {3,4}
                              {3,5}
                              {4,5}
                              {2,3,5}
                              {3,4,5}
a(n) is also the number of subsets of {1..n} containing all of their pairwise products <= n as well as any quotients of divisible elements. For example, the a(0) = 1 through a(5) = 13 subsets are:
  {}  {}   {}     {}       {}         {}
      {1}  {1}    {1}      {1}        {1}
           {1,2}  {1,2}    {1,3}      {1,3}
                  {1,3}    {1,4}      {1,4}
                  {1,2,3}  {1,2,4}    {1,5}
                           {1,3,4}    {1,2,4}
                           {1,2,3,4}  {1,3,4}
                                      {1,3,5}
                                      {1,4,5}
                                      {1,2,3,4}
                                      {1,2,4,5}
                                      {1,3,4,5}
                                      {1,2,3,4,5}
Also the number of subsets of {1..n} containing all of their multiples <= n. For example, the a(0) = 1 through a(5) = 13 subsets are:
  {}  {}   {}     {}       {}         {}
      {1}  {2}    {2}      {3}        {3}
           {1,2}  {3}      {4}        {4}
                  {2,3}    {2,4}      {5}
                  {1,2,3}  {3,4}      {2,4}
                           {2,3,4}    {3,4}
                           {1,2,3,4}  {3,5}
                                      {4,5}
                                      {2,3,4}
                                      {2,4,5}
                                      {3,4,5}
                                      {2,3,4,5}
                                      {1,2,3,4,5}
(End)
From _Gus Wiseman_, Mar 12 2024: (Start)
Also the number of subsets of {1..n} containing all divisors of the elements. For example, the a(0) = 1 through a(6) = 17 subsets are:
  {}  {}   {}     {}       {}         {}
      {1}  {1}    {1}      {1}        {1}
           {1,2}  {1,2}    {1,2}      {1,2}
                  {1,3}    {1,3}      {1,3}
                  {1,2,3}  {1,2,3}    {1,5}
                           {1,2,4}    {1,2,3}
                           {1,2,3,4}  {1,2,4}
                                      {1,2,5}
                                      {1,3,5}
                                      {1,2,3,4}
                                      {1,2,3,5}
                                      {1,2,4,5}
                                      {1,2,3,4,5}
(End)

References

  • Blanchet-Sadri, Francine. Algorithmic combinatorics on partial words. Chapman & Hall/CRC, Boca Raton, FL, 2008. ii+385 pp. ISBN: 978-1-4200-6092-8; 1-4200-6092-9 MR2384993 (2009f:68142). See p. 320. - N. J. A. Sloane, Apr 06 2012

Links

Crossrefs

Cf. A007865, A054519, A096827, A103580, A303362, A305148.
Cf. A326023, A326076, A326077, A326081, A326082, A326083, A326117.
Cf. A037031, A051026, A355740, A368110, A370585.

Programs

  • Maple
    with(numtheory):
b:= proc(s) option remember; local n;
      n:= max(s[]);
      `if`(n<0, 1, b(s minus {n}) + b(s minus divisors(n)))
    end:
bb:= n-> b({$2..n} minus divisors(n)):
sb:= proc(n) option remember; `if`(n<2, 0, bb(n) + sb(n-1)) end:
a:= n-> `if`(n=0, 1, `if`(isprime(n), 2*a(n-1)-1, 2+sb(n))):
seq(a(n), n=0..40);  # Alois P. Heinz, Mar 07 2011
  • Mathematica
    b[s_] := b[s] = With[{n=Max[s]}, If[n < 0, 1, b[Complement[s, {n}]] + b[Complement[s, Divisors[n]]]]];
bb[n_] := b[Complement[Range[2, n], Divisors[n]]];
sb[n_] := sb[n] = If[n < 2, 0, bb[n] + sb[n-1]];
a[n_] := If[n == 0, 1, If[PrimeQ[n], 2a[n-1] - 1, 2 + sb[n]]]; Table[a[n], {n, 0, 37}]
(* Jean-François Alcover, Jul 27 2011, converted from Maple *)
Table[Length[Select[Subsets[Range[n]], SubsetQ[#,Select[Union@@Table[#*i,{i,n}],#<=n&]]&]],{n,10}] (* Gus Wiseman, Jun 07 2019 *)
Table[Length[Select[Subsets[Range[n]], #==Union@@Divisors/@#&]],{n,0,10}] (* Gus Wiseman, Mar 12 2024 *)

Extensions

More terms from David Wasserman, May 02 2002
a(32)-a(37) from Donovan Johnson, Aug 11 2010

A370592 Number of integer partitions of n such that it is possible to choose a different prime factor of each part.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 1, 3, 3, 4, 4, 5, 6, 7, 9, 11, 12, 12, 16, 18, 22, 26, 29, 29, 37, 41, 49, 55, 61, 68, 72, 88, 98, 110, 120, 135, 146, 166, 190, 209, 227, 252, 277, 309, 346, 379, 413, 447, 500, 548, 606, 665, 727, 785, 857, 949, 1033, 1132, 1228, 1328, 1440
Offset: 0

Views

Author

Gus Wiseman, Feb 29 2024

Keywords

Examples

			The partition (10,6,4) has choice (5,3,2) so is counted under a(20).
The a(0) = 1 through a(10) = 4 partitions:
  ()  .  (2)  (3)  (4)  (5)    (6)  (7)    (8)    (9)    (10)
                        (3,2)       (4,3)  (5,3)  (5,4)  (6,4)
                                    (5,2)  (6,2)  (6,3)  (7,3)
                                                  (7,2)  (5,3,2)
The a(0) = 1 through a(17) = 12 partitions (0 = {}, A..H = 10..17):
  0  .  2  3  4  5   6  7   8   9   A    B   C    D    E    F    G    H
                 32     43  53  54  64   65  66   76   86   87   97   98
                        52  62  63  73   74  75   85   95   96   A6   A7
                                72  532  83  A2   94   A4   A5   B5   B6
                                         92  543  A3   B3   B4   C4   C5
                                             732  B2   C2   C3   D3   D4
                                                  652  653  D2   E2   E3
                                                       743  654  754  F2
                                                       752  753  763  665
                                                            762  853  764
                                                            A32  952  A43
                                                                 B32  7532

Crossrefs

The version for divisors instead of factors is A239312, ranks A368110.
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.
These partitions have ranks A368100.
The version for factorizations is A368414, complement A368413.
The complement is counted by A370593, ranks A355529.
For a unique choice we have A370594, ranks A370647.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A355741 counts choices of a prime factor of each prime index.
Cf. A000040, A000720, A133686, A355739, A355740, A355745, A367771, A367905, A370585, A370586, A370636.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], Length[Select[Tuples[If[#==1, {},First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]>0&]],{n,0,30}]

Formula

a(n) = A000041(n) - A370593(n).

A370593 Number of integer partitions of n such that it is not possible to choose a different prime factor of each part.

Original entry on oeis.org

0, 1, 1, 2, 4, 5, 10, 12, 19, 26, 38, 51, 71, 94, 126, 165, 219, 285, 369, 472, 605, 766, 973, 1226, 1538, 1917, 2387, 2955, 3657, 4497, 5532, 6754, 8251, 10033, 12190, 14748, 17831, 21471, 25825, 30976, 37111, 44331, 52897, 62952, 74829, 88755, 105145, 124307
Offset: 0

Views

Author

Gus Wiseman, Feb 29 2024

Keywords

Examples

			The a(0) = 0 through a(7) = 12 partitions:
  .  (1)  (11)  (21)   (22)    (41)     (33)      (61)
                (111)  (31)    (221)    (42)      (322)
                       (211)   (311)    (51)      (331)
                       (1111)  (2111)   (222)     (421)
                               (11111)  (321)     (511)
                                        (411)     (2221)
                                        (2211)    (3211)
                                        (3111)    (4111)
                                        (21111)   (22111)
                                        (111111)  (31111)
                                                  (211111)
                                                  (1111111)

Crossrefs

The complement for divisors instead of factors is A239312, ranks A368110.
These partitions have ranks A355529, complement A368100.
The complement for set-systems is A367902, ranks A367906, unlabeled A368095.
The version for set-systems is A367903, ranks A367907, unlabeled A368094.
For unlabeled multiset partitions we have A368097, complement A368098.
The version for factorizations is A368413, complement A368414.
The complement is counted by A370592.
For a unique choice we have A370594, ranks A370647.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A355741 counts choices of a prime factor of each prime index.
Cf. A000040, A000720, A133686, A355739, A355740, A367771, A367867, A367905, A370583, A370585, A370586, A370636.

Programs

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

Formula

a(n) = A000041(n) - A370592(n).

A370808 Greatest number of multisets that can be obtained by choosing a divisor of each part of an integer partition of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 6, 7, 10, 11, 14, 17, 19, 23, 29, 30, 39, 41, 51, 58, 66, 78, 82, 102, 110, 132, 144, 162, 186, 210, 228, 260, 296, 328, 366, 412, 462, 512, 560, 638, 692, 764, 860, 924, 1028, 1122, 1276, 1406, 1528, 1721, 1898, 2056, 2318, 2506, 2812, 3020, 3442
Offset: 0

Views

Author

Gus Wiseman, Mar 05 2024

Keywords

Examples

			For the partitions of 5 we have the following choices:
      (5): {{1},{5}}
     (41): {{1,1},{1,2},{1,4}}
     (32): {{1,1},{1,2},{1,3},{2,3}}
    (311): {{1,1,1},{1,1,3}}
    (221): {{1,1,1},{1,1,2},{1,2,2}}
   (2111): {{1,1,1,1},{1,1,1,2}}
  (11111): {{1,1,1,1,1}}
So a(5) = 4.

Crossrefs

For just prime factors we have A370809.
The version for factorizations is A370816, for just prime factors A370817.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A239312 counts condensed partitions, ranks A368110.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355733 counts choices of divisors of prime indicec.
A370320 counts non-condensed partitions, ranks A355740.
A370592 counts factor-choosable partitions, complement A370593.
Cf. A000792, A048249, A066739, A319055, A355737, A355739, A370348, A370595, A370803, A370810.

Programs

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

Extensions

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

A370813 Number of non-condensed integer factorizations of n into unordered factors > 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Gus Wiseman, Mar 04 2024

Keywords

Comments

A multiset is condensed iff it is possible to choose a different divisor of each element.

Examples

			The a(96) = 4 factorizations: (2*2*2*2*2*3), (2*2*2*2*6), (2*2*2*3*4), (2*2*2*12).

Crossrefs

Partitions not of this type are counted by A239312, ranks A368110.
Factors instead of divisors: A368413, complement A368414, unique A370645.
Partitions of this type are counted by A370320, ranks A355740.
Subsets of this type: A370583 and A370637, complement A370582 and A370636.
The complement is counted by A370814, partitions A370592, ranks A368100.
For a unique choice we have A370815, partitions A370595, ranks A370810.
A000005 counts divisors.
A001055 counts factorizations, strict A045778.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Cf. A340596, A340653, A355529, A355739, A355741, A370804, A370807.

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[Select[Tuples[Divisors /@ #],UnsameQ@@#&]]==0&]],{n,100}]

A370320 Number of non-condensed integer partitions of n, or partitions where it is not possible to choose a different divisor of each part.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 6, 9, 13, 20, 28, 40, 54, 74, 102, 135, 180, 235, 310, 397, 516, 658, 843, 1066, 1349, 1687, 2119, 2634, 3273, 4045, 4995, 6128, 7517, 9171, 11181, 13579, 16457, 19884, 23992, 28859, 34646, 41506, 49634, 59211, 70533, 83836, 99504, 117867
Offset: 0

Views

Author

Gus Wiseman, Mar 02 2024

Keywords

Comments

Includes all partitions containing 1.

Examples

			The a(0) = 0 through a(8) = 13 partitions:
  .  .  (11)  (111)  (211)   (221)    (222)     (331)      (611)
                     (1111)  (311)    (411)     (511)      (2222)
                             (2111)   (2211)    (2221)     (3221)
                             (11111)  (3111)    (3211)     (3311)
                                      (21111)   (4111)     (4211)
                                      (111111)  (22111)    (5111)
                                                (31111)    (22211)
                                                (211111)   (32111)
                                                (1111111)  (41111)
                                                           (221111)
                                                           (311111)
                                                           (2111111)
                                                           (11111111)

Crossrefs

The complement is counted by A239312 (condensed partitions).
These partitions have ranks A355740.
Factorizations in the case of prime factors are A368413, complement A368414.
The complement for prime factors is A370592, ranks A368100.
The version for prime factors (not all divisors) is A370593, ranks A355529.
For a unique choice we have A370595, ranks A370810.
For multiple choices we have A370803, ranks A370811.
The case without ones is A370804, complement A370805.
The version for factorizations is A370813, complement A370814.
A000005 counts divisors.
A000041 counts integer partitions.
A027746 lists prime factors, A112798 indices, length A001222.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355741 chooses prime factors of prime indices, variations A355744, A355745.
Cf. A355535, A355739, A367867, A368097, A368110, A370583, A370584, A370594, A370806, A370807, A370808.

Programs

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

Extensions

a(31)-a(47) from Alois P. Heinz, Mar 03 2024

A370814 Number of condensed integer factorizations of n into unordered factors > 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 2, 4, 1, 5, 1, 5, 2, 2, 2, 8, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 10, 2, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 11, 1, 2, 4, 7, 2, 5, 1, 4, 2, 5, 1, 14, 1, 2, 4, 4, 2, 5, 1, 10, 4, 2, 1, 11, 2
Offset: 1

Views

Author

Gus Wiseman, Mar 04 2024

Keywords

Comments

A multiset is condensed iff it is possible to choose a different divisor of each element.

Examples

			The a(36) = 7 factorizations: (2*2*9), (2*3*6), (2*18), (3*3*4), (3*12), (4*9), (6*6), (36).

Crossrefs

Partitions of this type are counted by A239312, ranks A368110.
Factors instead of divisors: A368414, complement A368413, unique A370645.
Partitions not of this type are counted by A370320, ranks A355740.
Subsets of this type: A370582 and A370636, complement A370583 and A370637.
The complement is counted by A370813, partitions A370593, ranks A355529.
For a unique choice we have A370815, partitions A370595, ranks A370810.
A000005 counts divisors.
A001055 counts factorizations, strict A045778.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Cf. A340596, A340653, A355739, A370592, A370803, A370804, A370805.

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[Select[Tuples[Divisors /@ #],UnsameQ@@#&]]>0&]],{n,100}]

A370583 Number of subsets of {1..n} such that it is not possible to choose a different prime factor of each element.

Original entry on oeis.org

0, 1, 2, 4, 10, 20, 44, 88, 204, 440, 908, 1816, 3776, 7552, 15364, 31240, 63744, 127488, 257592, 515184, 1036336, 2079312, 4166408, 8332816, 16709632, 33470464, 66978208, 134067488, 268236928, 536473856, 1073233840, 2146467680, 4293851680, 8588355424, 17177430640
Offset: 0

Views

Author

Gus Wiseman, Feb 26 2024

Keywords

Examples

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

Crossrefs

Multisets of this type are ranked by A355529, complement A368100.
For divisors instead of factors we have A355740, complement A368110.
The complement for set-systems is A367902, ranks A367906, unlabeled A368095.
The version for set-systems is A367903, ranks A367907, unlabeled A368094.
For non-isomorphic multiset partitions we have A368097, complement A368098.
The version for factorizations is A368413, complement A368414.
The complement is counted by A370582.
For a unique choice we have A370584.
Partial sums of A370587, complement A370586.
The minimal case is A370591.
The version for partitions is A370593, complement A370592.
For binary indices instead of factors we have A370637, complement A370636.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A355741 counts choices of a prime factor of each prime index.
Cf. A000040, A000720, A001055, A001414, A003963, A005117, A045778, A355739, A355745, A367867, A367905, A368187.

Programs

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

Formula

a(n) = 2^n - A370582(n).

Extensions

a(19)-a(34) from Alois P. Heinz, Feb 27 2024
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