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 1-5 of 5 results.

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

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

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
Showing 1-5 of 5 results.

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