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 11 results. Next

A342495 Number of compositions of n with constant (equal) first quotients.

Original entry on oeis.org

1, 1, 2, 4, 5, 6, 8, 10, 10, 11, 12, 12, 16, 16, 18, 20, 19, 18, 22, 22, 24, 28, 24, 24, 30, 27, 30, 30, 34, 30, 38, 36, 36, 36, 36, 40, 43, 40, 42, 46, 48, 42, 52, 46, 48, 52, 48, 48, 56, 55, 54, 54, 58, 54, 60, 58, 64, 64, 60, 60, 72, 64, 68, 74, 69, 72, 72
Offset: 0

Views

Author

Gus Wiseman, Mar 17 2021

Keywords

Comments

The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

Examples

			The composition (1,2,4,8) has first quotients (2,2,2) so is counted under a(15).
The composition (4,5,6) has first quotients (5/4,6/5) so is not counted under a(15).
The a(1) = 1 through a(7) = 10 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (12)   (13)    (14)     (15)      (16)
             (21)   (22)    (23)     (24)      (25)
             (111)  (31)    (32)     (33)      (34)
                    (1111)  (41)     (42)      (43)
                            (11111)  (51)      (52)
                                     (222)     (61)
                                     (111111)  (124)
                                               (421)
                                               (1111111)

Links

Crossrefs

The version for differences instead of quotients is A175342.
The unordered version is A342496, ranked by A342522.
The strict unordered version is A342515.
The distinct version is A342529.
A000005 counts constant compositions.
A000009 counts strictly increasing (or strictly decreasing) compositions.
A000041 counts weakly increasing (or weakly decreasing) compositions.
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A167865 counts strict chains of divisors > 1 summing to n.
Cf. A002843, A003242, A008965, A048004, A059966, A074206, A167606, A253249, A318991, A318992, A325557, A342528.

Programs

  • Mathematica
    Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ@@Divide@@@Partition[#,2,1]&]],{n,0,15}]

Formula

a(n > 0) = 2*A342496(n) - A000005(n).

A342496 Number of integer partitions of n with constant (equal) first quotients.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 6, 6, 7, 7, 8, 7, 11, 9, 11, 12, 12, 10, 14, 12, 15, 16, 14, 13, 19, 15, 17, 17, 20, 16, 23, 19, 21, 20, 20, 22, 26, 21, 23, 25, 28, 22, 30, 24, 27, 29, 26, 25, 33, 29, 30, 29, 32, 28, 34, 31, 36, 34, 32, 31, 42
Offset: 0

Views

Author

Gus Wiseman, Mar 17 2021

Keywords

Comments

The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

Examples

			The partition (12,6,3) has first quotients (1/2,1/2) so is counted under a(21).
The a(1) = 1 through a(9) = 7 partitions:
  1   2    3     4      5       6        7         8          9
      11   21    22     32      33       43        44         54
           111   31     41      42       52        53         63
                 1111   11111   51       61        62         72
                                222      421       71         81
                                111111   1111111   2222       333
                                                   11111111   111111111

Links

Crossrefs

The version for differences instead of quotients is A049988.
The ordered version is A342495.
The distinct version is A342514.
The strict case is A342515.
The Heinz numbers of these partitions are A342522.
A000005 counts constant partitions.
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A167865 counts strict chains of divisors > 1 summing to n.
Cf. A000837, A002843, A003242, A074206, A175342, A318991, A318992, A325557, A342527, A342528, A342529.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],SameQ@@Divide@@@Partition[#,2,1]&]],{n,0,30}]

Formula

a(n > 0) = (A342495(n) + A000005(n))/2.

A342514 Number of integer partitions of n with distinct first quotients.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 6, 8, 11, 14, 18, 24, 28, 35, 41, 52, 64, 81, 93, 115, 137, 157, 190, 225, 268, 313, 366, 430, 502, 587, 683, 790, 913, 1055, 1217, 1393, 1605, 1830, 2098, 2384, 2722, 3101, 3524, 4005, 4524, 5137, 5812, 6570, 7434, 8360, 9416, 10602, 11881
Offset: 0

Views

Author

Gus Wiseman, Mar 17 2021

Keywords

Comments

Also the number of reversed integer partitions of n with distinct first quotients.
The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

Examples

			The partition (4,3,3,2,1) has first quotients (3/4,1,2/3,1/2) so is counted under a(13), but it has first differences (-1,0,-1,-1) so is not counted under A325325(13).
The a(1) = 1 through a(9) = 14 partitions:
  (1)  (2)   (3)   (4)    (5)    (6)    (7)     (8)     (9)
       (11)  (21)  (22)   (32)   (33)   (43)    (44)    (54)
                   (31)   (41)   (42)   (52)    (53)    (63)
                   (211)  (221)  (51)   (61)    (62)    (72)
                          (311)  (321)  (322)   (71)    (81)
                                 (411)  (331)   (332)   (432)
                                        (511)   (422)   (441)
                                        (3211)  (431)   (522)
                                                (521)   (531)
                                                (611)   (621)
                                                (3221)  (711)
                                                        (3321)
                                                        (4311)
                                                        (5211)

Links

Crossrefs

The version for differences instead of quotients is A325325.
The ordered version is A342529.
The strict case is A342520.
The Heinz numbers of these partitions are A342521.
A000005 counts constant partitions.
A000009 counts strict partitions.
A000041 counts partitions.
A001055 counts factorizations (strict: A045778, ordered: A074206).
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A167865 counts strict chains of divisors > 1 summing to n.
A342096 counts partitions with all adjacent parts x < 2y (strict: A342097).
A342098 counts partitions with all adjacent parts x > 2y.
Cf. A000837, A002843, A003242, A175342, A318991, A318992, A325557, A342527, A342528, A342529.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@Divide@@@Partition[#,2,1]&]],{n,0,30}]

A342520 Number of strict integer partitions of n with distinct first quotients.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 4, 6, 8, 10, 12, 13, 16, 20, 25, 30, 37, 42, 50, 57, 65, 80, 93, 108, 127, 147, 170, 198, 225, 258, 297, 340, 385, 448, 499, 566, 647, 737, 832, 937, 1064, 1186, 1348, 1522, 1701, 1916, 2157, 2402, 2697, 3013, 3355, 3742, 4190, 4656, 5191
Offset: 0

Views

Author

Gus Wiseman, Mar 20 2021

Keywords

Comments

Also the number of reversed strict integer partitions of n with distinct first quotients.
The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

Examples

			The strict partition (12,10,5,2,1) has first quotients (5/6,1/2,2/5,1/2) so is not counted under a(30), even though the first differences (-2,-5,-3,-1) are distinct.
The a(1) = 1 through a(13) = 16 partitions (A..D = 10..13):
  1   2   3    4    5    6     7    8     9     A      B      C     D
          21   31   32   42    43   53    54    64     65     75    76
                    41   51    52   62    63    73     74     84    85
                         321   61   71    72    82     83     93    94
                                    431   81    91     92     A2    A3
                                    521   432   532    A1     B1    B2
                                          531   541    542    543   C1
                                          621   631    632    642   643
                                                721    641    651   652
                                                4321   731    732   742
                                                       821    741   751
                                                       5321   831   832
                                                              921   841
                                                                    A21
                                                                    5431
                                                                    7321

Links

Crossrefs

The version for differences instead of quotients is A320347.
The non-strict version is A342514 (ranking: A342521).
The equal instead of distinct version is A342515.
The non-strict ordered version is A342529.
The version for strict divisor chains is A342530.
A000041 counts partitions (strict: A000009).
A001055 counts factorizations (strict: A045778, ordered: A074206).
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A167865 counts strict chains of divisors > 1 summing to n.
A342086 counts strict chains of divisors with strictly increasing quotients.
A342098 counts (strict) partitions with all adjacent parts x > 2y.
Cf. A000005, A003114, A003242, A005117, A018819, A067824, A238710, A253249, A318991, A318992.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&UnsameQ@@Divide@@@Partition[#,2,1]&]],{n,0,30}]

A342530 Number of strict chains of divisors ending with n and having distinct first quotients.

Original entry on oeis.org

1, 2, 2, 3, 2, 6, 2, 6, 3, 6, 2, 12, 2, 6, 6, 9, 2, 12, 2, 12, 6, 6, 2, 28, 3, 6, 6, 12, 2, 26, 2, 14, 6, 6, 6, 31, 2, 6, 6, 28, 2, 26, 2, 12, 12, 6, 2, 52, 3, 12, 6, 12, 2, 28, 6, 28, 6, 6, 2, 66, 2, 6, 12, 25, 6, 26, 2, 12, 6, 26, 2, 76, 2, 6, 12, 12, 6, 26
Offset: 1

Views

Author

Gus Wiseman, Mar 25 2021

Keywords

Comments

The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the quotients of (6,3,1) are (1/2,1/3).

Examples

			The a(1) = 1 through a(12) = 12 chains (reversed):
  1  2    3    4    5    6      7    8      9    10      11    12
     2/1  3/1  4/1  5/1  6/1    7/1  8/1    9/1  10/1    11/1  12/1
               4/2       6/2         8/2    9/3  10/2          12/2
                         6/3         8/4         10/5          12/3
                         6/2/1       8/2/1       10/2/1        12/4
                         6/3/1       8/4/1       10/5/1        12/6
                                                               12/2/1
                                                               12/3/1
                                                               12/4/1
                                                               12/4/2
                                                               12/6/1
                                                               12/6/2
Not counted under a(12) are: 12/4/2/1, 12/6/2/1, 12/6/3, 12/6/3/1.

Links

Crossrefs

The version for weakly increasing first quotients is A057567.
The version for equal first quotients is A169594.
The case of chains starting with 1 is A254578.
The version for strictly increasing first quotients is A342086.
A001055 counts factorizations (strict: A045778, ordered: A074206).
A067824 counts strict chains of divisors ending with n.
A167865 counts strict chains of divisors > 1 summing to n.
A253249 counts strict chains of divisors.
A334997 counts chains of divisors of n by length.
A342495/A342529 count compositions with equal/distinct quotients.
A342496/A342514 count partitions with equal/distinct quotients.
A342515/A342520 count strict partitions with equal/distinct quotients.
A342522/A342521 rank partitions with equal/distinct quotients.
Cf. A000009, A003238, A003242, A122651, A179254, A318991, A318992, A325545.

Programs

  • Mathematica
    cmi[n_]:=Prepend[Prepend[#,n]&/@Join@@cmi/@Most[Divisors[n]],{n}];
Table[Length[Select[cmi[n],UnsameQ@@Divide@@@Partition[#,2,1]&]],{n,100}]

Formula

a(n) = Sum_{d|n} A254578(d). - Ridouane Oudra, Jun 17 2025

A342532 Number of even-length compositions of n with alternating parts distinct.

Original entry on oeis.org

1, 0, 1, 2, 3, 4, 9, 14, 28, 44, 83, 136, 250, 424, 757, 1310, 2313, 4018, 7081, 12314, 21650, 37786, 66264, 115802, 202950, 354858, 621525, 1087252, 1903668, 3330882, 5831192, 10204250, 17862232, 31260222, 54716913, 95762576, 167614445, 293356422, 513456686
Offset: 0

Views

Author

Gus Wiseman, Mar 28 2021

Keywords

Comments

These are finite even-length sequences q of positive integers summing to n such that q(i) != q(i+2) for all possible i.

Examples

			The a(2) = 1 through a(7) = 14 compositions:
  (1,1)  (1,2)  (1,3)  (1,4)  (1,5)      (1,6)
         (2,1)  (2,2)  (2,3)  (2,4)      (2,5)
                (3,1)  (3,2)  (3,3)      (3,4)
                       (4,1)  (4,2)      (4,3)
                              (5,1)      (5,2)
                              (1,1,2,2)  (6,1)
                              (1,2,2,1)  (1,1,2,3)
                              (2,1,1,2)  (1,1,3,2)
                              (2,2,1,1)  (1,2,3,1)
                                         (1,3,2,1)
                                         (2,1,1,3)
                                         (2,3,1,1)
                                         (3,1,1,2)
                                         (3,2,1,1)

Links

Crossrefs

The strictly decreasing version appears to be A064428 (odd-length: A001522).
The equal version is A065608 (A342527 with odds).
The weakly decreasing version is A114921 (A342528 with odds).
Including odds gives A224958.
A000726 counts partitions with alternating parts unequal.
A325545 counts compositions with distinct first differences.
A342529 counts compositions with distinct first quotients.
Cf. A000009, A000041, A003242, A008965, A032020, A059966, A062968, A064410, A070211, A106351, A175342, A325546.

Programs

  • Mathematica
    qdq[q_]:=And@@Table[q[[i]]!=q[[i+2]],{i,Length[q]-2}];
Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],EvenQ[Length[#]]&],qdq]],{n,0,15}]
  • PARI
    \\ here gf gives A106351 as g.f.
gf(n, y)={1/(1 - sum(k=1, n, (-1)^(k+1)*x^k*y^k/(1-x^k) + O(x*x^n)))}
seq(n)={my(p=gf(n,y)); Vec(sum(k=0, n\2, polcoef(p,k,y)^2))} \\ Andrew Howroyd, Apr 16 2021

Formula

G.f.: 1 + Sum_{k>=1} B_k(x)^2 where B_k(x) is the g.f. of column k of A106351. - Andrew Howroyd, Apr 16 2021

Extensions

Terms a(24) and beyond from Andrew Howroyd, Apr 16 2021

A342498 Number of integer partitions of n with strictly increasing first quotients.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 5, 6, 8, 9, 12, 12, 14, 16, 18, 20, 24, 26, 27, 30, 35, 37, 45, 47, 52, 56, 61, 65, 72, 77, 83, 90, 95, 99, 109, 117, 127, 135, 144, 151, 164, 172, 181, 197, 209, 222, 239, 249, 263, 280, 297, 310, 332, 349, 368, 391, 412, 433, 457, 480, 503
Offset: 0

Views

Author

Gus Wiseman, Mar 17 2021

Keywords

Comments

Also the number of reversed integer partitions of n with strictly increasing first quotients.
The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

Examples

			The partition y = (13,7,2,1) has first quotients (7/13,2/7,1/2) so is not counted under a(23). However, the first differences (-6,-5,-1) are strictly increasing, so y is counted under A240027(23).
The a(1) = 1 through a(9) = 9 partitions:
  (1)  (2)   (3)   (4)    (5)    (6)    (7)    (8)    (9)
       (11)  (21)  (22)   (32)   (33)   (43)   (44)   (54)
                   (31)   (41)   (42)   (52)   (53)   (63)
                   (211)  (311)  (51)   (61)   (62)   (72)
                                 (411)  (322)  (71)   (81)
                                        (511)  (422)  (522)
                                               (521)  (621)
                                               (611)  (711)
                                                      (5211)

Links

Crossrefs

The version for differences instead of quotients is A240027.
The ordered version is A342493.
The weakly increasing version is A342497.
The strictly decreasing version is A342499.
The strict case is A342517.
The Heinz numbers of these partitions are A342524.
A000005 counts constant partitions.
A000009 counts strict partitions.
A000041 counts partitions.
A001055 counts factorizations.
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A074206 counts ordered factorizations.
A167865 counts strict chains of divisors > 1 summing to n.
A342096 counts partitions with adjacent x < 2y (strict: A342097).
A342098 counts partitions with adjacent parts x > 2y.
Cf. A000837, A002843, A003242, A175342, A318991, A318992, A325557, A342527, A342528, A342529.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Less@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,30}]

A342499 Number of integer partitions of n with strictly decreasing first quotients.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 5, 7, 9, 10, 11, 14, 15, 18, 20, 23, 26, 31, 34, 39, 42, 45, 51, 58, 65, 70, 78, 83, 91, 102, 111, 122, 133, 145, 158, 170, 182, 202, 217, 231, 248, 268, 285, 307, 332, 354, 374, 404, 436, 468, 502, 537, 576, 618, 654, 694, 737, 782, 830
Offset: 0

Views

Author

Gus Wiseman, Mar 17 2021

Keywords

Comments

Also the number of reversed partitions of n with strictly decreasing first quotients.
The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

Examples

			The partition (6,6,3,1) has first quotients (1,1/2,1/3) so is counted under a(16).
The a(1) = 1 through a(9) = 9 partitions:
  (1)  (2)   (3)   (4)   (5)    (6)    (7)    (8)    (9)
       (11)  (21)  (22)  (32)   (33)   (43)   (44)   (54)
                   (31)  (41)   (42)   (52)   (53)   (63)
                         (221)  (51)   (61)   (62)   (72)
                                (321)  (331)  (71)   (81)
                                              (332)  (432)
                                              (431)  (441)
                                                     (531)
                                                     (3321)

Links

Crossrefs

The version for differences instead of quotients is A320470.
The ordered version is A342494.
The strictly increasing version is A342498.
The weakly decreasing version is A342513.
The strict case is A342518.
The Heinz numbers of these partitions are listed by A342525.
A000005 counts constant partitions.
A000009 counts strict partitions.
A000041 counts partitions.
A001055 counts factorizations.
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A074206 counts ordered factorizations.
A167865 counts strict chains of divisors > 1 summing to n.
A342096 counts partitions with adjacent x < 2y (strict: A342097).
A342098 counts partitions with adjacent parts x > 2y.
Cf. A000837, A002843, A003242, A175342, A318991, A318992, A325557, A342527, A342528, A342529.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Greater@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,30}]

A342521 Heinz numbers of integer partitions with distinct first quotients.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82
Offset: 1

Views

Author

Gus Wiseman, Mar 23 2021

Keywords

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

Examples

			The prime indices of 1365 are {2,3,4,6}, with first quotients (3/2,4/3,3/2), so 1365 is not in the sequence.
Most small numbers are in the sequence, but the sequence of non-terms together with their prime indices begins:
    8: {1,1,1}
   16: {1,1,1,1}
   24: {1,1,1,2}
   27: {2,2,2}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   48: {1,1,1,1,2}
   54: {1,2,2,2}
   56: {1,1,1,4}
   64: {1,1,1,1,1,1}
   72: {1,1,1,2,2}
   80: {1,1,1,1,3}
   81: {2,2,2,2}
   84: {1,1,2,4}
   88: {1,1,1,5}
   96: {1,1,1,1,1,2}
  100: {1,1,3,3}

Links

Crossrefs

For multiplicities (prime signature) instead of quotients we have A130091.
For differences instead of quotients we have A325368 (count: A325325).
These partitions are counted by A342514 (strict: A342520, ordered: A342529).
The equal instead of distinct version is A342522.
The version counting strict divisor chains is A342530.
A001055 counts factorizations (strict: A045778, ordered: A074206).
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A167865 counts strict chains of divisors > 1 summing to n.
A318991/A318992 rank reversed partitions with/without integer quotients.
Cf. A003242, A005117, A056239, A067824, A098859, A112798, A169594, A253249, A325326, A325337, A325405.

Programs

  • Mathematica
    primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],UnsameQ@@Divide@@@Reverse/@Partition[primeptn[#],2,1]&]

A342497 Number of integer partitions of n with weakly increasing first quotients.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 9, 11, 15, 18, 23, 25, 32, 36, 43, 49, 60, 65, 75, 83, 96, 106, 121, 131, 150, 163, 178, 194, 217, 230, 254, 275, 300, 320, 350, 374, 411, 439, 470, 503, 548, 578, 625, 666, 710, 758, 815, 855, 913, 970, 1029, 1085, 1157, 1212, 1288, 1360
Offset: 0

Views

Author

Gus Wiseman, Mar 17 2021

Keywords

Comments

Also called log-concave-up partitions.
Also the number of reversed integer partitions of n with weakly increasing first quotients.
The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

Examples

			The partition y = (6,3,2,1,1) has first quotients (1/2,2/3,1/2,1) so is not counted under a(13). However, the first differences (-3,-1,-1,0) are weakly increasing, so y is counted under A240026(13).
The a(1) = 1 through a(8) = 15 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (211)   (311)    (51)      (61)       (62)
                    (1111)  (2111)   (222)     (322)      (71)
                            (11111)  (411)     (421)      (422)
                                     (3111)    (511)      (521)
                                     (21111)   (4111)     (611)
                                     (111111)  (31111)    (2222)
                                               (211111)   (4211)
                                               (1111111)  (5111)
                                                          (41111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

Links

Crossrefs

The version for differences instead of quotients is A240026.
The ordered version is A342492.
The strictly increasing version is A342498.
The weakly decreasing version is A342513.
The strict case is A342516.
The Heinz numbers of these partitions are A342523.
A000005 counts constant partitions.
A000009 counts strict partitions.
A000041 counts partitions.
A000929 counts partitions with all adjacent parts x >= 2y.
A001055 counts factorizations.
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A074206 counts ordered factorizations.
A167865 counts strict chains of divisors > 1 summing to n.
A342094 counts partitions with all adjacent parts x <= 2y.
Cf. A000837, A002843, A003242, A175342, A318991, A318992, A325557, A342527, A342528, A342529.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],LessEqual@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,30}]
Showing 1-10 of 11 results. Next

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