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 1-4 of 4 results.

A342493 Number of compositions of n with strictly increasing first quotients.

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 11, 16, 22, 28, 39, 49, 61, 77, 93, 114, 140, 169, 198, 233, 276, 321, 381, 439, 509, 591, 678, 774, 883, 1007, 1147, 1300, 1465, 1641, 1845, 2068, 2317, 2590, 2881, 3193, 3549, 3928, 4341, 4793, 5282, 5813, 6401, 7027, 7699, 8432, 9221, 10076
Offset: 0

Views

Author

Gus Wiseman, Mar 16 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 (3,1,1,2) has first quotients (1/3,1,2) so is counted under a(7).
The a(1) = 1 through a(7) = 16 compositions:
  (1)  (2)    (3)    (4)      (5)      (6)        (7)
       (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)
                     (1,1,2)  (4,1)    (4,2)      (4,3)
                     (2,1,1)  (1,1,3)  (5,1)      (5,2)
                              (2,1,2)  (1,1,4)    (6,1)
                              (3,1,1)  (2,1,3)    (1,1,5)
                                       (3,1,2)    (2,1,4)
                                       (4,1,1)    (2,2,3)
                                       (2,1,1,2)  (3,1,3)
                                                  (3,2,2)
                                                  (4,1,2)
                                                  (5,1,1)
                                                  (2,1,1,3)
                                                  (3,1,1,2)

Links

Crossrefs

The version for differences instead of quotients is A325547.
The weakly increasing version is A342492.
The strictly decreasing version is A342494.
The unordered version is A342498, ranked by A342524.
The strict unordered version is A342517.
A000005 counts constant compositions.
A000009 counts strictly increasing (or strictly decreasing) compositions.
A000041 counts weakly increasing (or weakly decreasing) compositions.
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.
A274199 counts compositions with all adjacent parts x < 2y.
Cf. A003242, A008965, A048004, A059966, A067824, A167606, A253249, A318991, A318992, A342527, A342528.

Programs

  • Maple
    b:= proc(n, q, l) option remember; `if`(n=0, 1, add(
     `if`(q=0 or q>l/j, b(n-j, l/j, j), 0), j=1..n))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..55);  # Alois P. Heinz, Mar 25 2021
  • Mathematica
    Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Less@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,15}]
(* Second program: *)
b[n_, q_, l_] := b[n, q, l] = If[n == 0, 1, Sum[
     If[q == 0 || q > l/j, b[n - j, l/j, j], 0], {j, 1, n}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 55] (* Jean-François Alcover, May 19 2021, after Alois P. Heinz *)

Extensions

a(21)-a(51) from Alois P. Heinz, Mar 18 2021

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}]

A342518 Number of strict integer partitions of n with strictly decreasing first quotients.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 4, 5, 7, 8, 9, 11, 12, 13, 17, 18, 21, 24, 28, 30, 34, 37, 41, 47, 52, 56, 63, 68, 72, 83, 89, 99, 108, 117, 128, 139, 149, 163, 179, 189, 203, 217, 233, 250, 272, 289, 305, 329, 355, 381, 410, 438, 471, 505, 540, 571, 607, 645, 683, 726
Offset: 0

Views

Author

Gus Wiseman, Mar 20 2021

Keywords

Comments

Also the number of reversed strict integer 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 strict partition (12,10,6,3,1) has first quotients (5/6,3/5,1/2,1/3) so is counted under a(32), even though the differences (-2,-4,-3,-2) are not strictly decreasing.
The a(1) = 1 through a(13) = 12 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
                                          432   541    A1    B1    B2
                                          531   631    542   543   C1
                                                4321   641   642   652
                                                       731   651   742
                                                             741   751
                                                             831   841
                                                                   5431

Links

Crossrefs

The version for differences instead of quotients is A320388.
The version for chains of divisors is A342086 (non-strict: A057567).
The non-strict ordered version is A342494.
The non-strict version is A342499 (ranking: A342525).
The strictly increasing version is A342517.
The weakly decreasing version is A342519.
A000041 counts partitions (strict: A000009).
A001055 counts factorizations (strict: A045778, ordered: A074206).
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A045690 counts sets with maximum n with all adjacent elements y < 2x.
A167865 counts strict chains of divisors > 1 summing to n.
A342096 counts partitions with all adjacent parts x < 2y (strict: A342097).
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@@#&&Greater@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,30}]

A342525 Heinz numbers of integer partitions with strictly decreasing first quotients.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 70, 71, 73, 74, 75, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98
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 150 are {1,2,3,3}, with first quotients (2,3/2,1), so 150 is 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}
   12: {1,1,2}
   16: {1,1,1,1}
   20: {1,1,3}
   24: {1,1,1,2}
   27: {2,2,2}
   28: {1,1,4}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}

Links

Crossrefs

For multiplicities (prime signature) instead of quotients we have A304686.
For differences instead of quotients we have A325457 (count: A320470).
The version counting strict divisor chains is A342086.
These partitions are counted by A342499 (strict: A342518, ordered: A342494).
The strictly increasing version is A342524.
The weakly decreasing version is A342526.
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.
A342098 counts (strict) partitions with all adjacent parts x > 2y.
Cf. A056239, A067824, A112798, A124010, A130091, A169594, A253249, A325351, A325352, A325405, A334997, A342530.

Programs

  • Mathematica
    primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],Greater@@Divide@@@Reverse/@Partition[primeptn[#],2,1]&]
Showing 1-4 of 4 results.

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