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.

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

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}]

A342516 Number of strict integer partitions of n with weakly increasing first quotients.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 5, 5, 6, 7, 8, 8, 11, 12, 14, 15, 17, 17, 21, 22, 26, 29, 31, 32, 35, 38, 42, 45, 48, 51, 58, 59, 63, 70, 76, 80, 88, 94, 98, 105, 113, 121, 129, 133, 143, 153, 159, 166, 183, 189, 195, 210, 221, 231, 248, 262, 273, 284, 298, 312
Offset: 0

Views

Author

Gus Wiseman, Mar 20 2021

Keywords

Comments

Also called log-concave-up strict partitions.
Also the number of reversed strict 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 (6,3,2,1) has first quotients (1/2,2/3,1/2) so is not counted under a(12), even though the first differences (-3,-1,-1) are weakly increasing.
The a(1) = 1 through a(13) = 11 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
                              61    71    72    82    83    93    94
                              421   521   81    91    92    A2    A3
                                          621   532   A1    B1    B2
                                                721   632   732   C1
                                                      821   921   643
                                                                  832
                                                                  931
                                                                  A21

Links

Crossrefs

The version for differences instead of quotients is A179255.
The non-strict ordered version is A342492.
The non-strict version is A342497 (ranking: A342523).
The strictly increasing version is A342517.
The weakly decreasing version is A342519.
A000041 counts partitions (strict: A000009).
A000929 counts partitions with all adjacent parts x >= 2y.
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.
A342094 counts partitions with all adjacent parts x <= 2y (strict: A342095).
Cf. A000005, A003114, A003242, A005117, A057567, A067824, A238710, A253249, A318991, A318992, A342528.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&LessEqual@@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}]

A342524 Heinz numbers of integer partitions with strictly increasing first quotients.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 71, 73, 74, 76, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91
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 84 are {1,1,2,4}, with first quotients (1,2,2), so 84 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}
   18: {1,2,2}
   24: {1,1,1,2}
   27: {2,2,2}
   30: {1,2,3}
   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}
   50: {1,3,3}
   54: {1,2,2,2}
   56: {1,1,1,4}
   60: {1,1,2,3}
   64: {1,1,1,1,1,1}

Links

Crossrefs

For differences instead of quotients we have A325456 (count: A240027).
For multiplicities (prime signature) instead of quotients we have A334965.
The version counting strict divisor chains is A342086.
These partitions are counted by A342498 (strict: A342517, ordered: A342493).
The weakly increasing version is A342523.
The strictly decreasing version is A342525.
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. A048767, A056239, A112798, A124010, A130091, A169594, A253249, A325351, A325352, A334997, A342530.

Programs

  • Mathematica
    primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],Less@@Divide@@@Reverse/@Partition[primeptn[#],2,1]&]
Showing 1-5 of 5 results.

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