A318992 -id:A318992 - cp's OEIS frontend

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

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

Gus Wiseman, Mar 17 2021

nonn

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).

			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)

Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

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.

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

A342492 Number of compositions of n with weakly increasing first quotients.

Original entry on oeis.org

1, 1, 2, 4, 7, 11, 17, 26, 37, 52, 73, 95, 125, 163, 208, 261, 330, 407, 498, 607, 734, 881, 1056, 1250, 1480, 1738, 2029, 2359, 2742, 3160, 3635, 4169, 4760, 5414, 6151, 6957, 7861, 8858, 9952, 11148, 12483, 13934, 15526, 17267, 19173, 21252, 23535, 25991

Offset: 0

Gus Wiseman, Mar 16 2021

nonn

Also called log-concave-up compositions.

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).

			The composition (4,2,1,2,3) has first quotients (1/2,1/2,2,3/2) so is not counted under a(12), even though the first differences (-2,-1,1,1) are weakly increasing.
The a(1) = 1 through a(6) = 17 compositions:
  (1)  (2)    (3)      (4)        (5)          (6)
       (1,1)  (1,2)    (1,3)      (1,4)        (1,5)
              (2,1)    (2,2)      (2,3)        (2,4)
              (1,1,1)  (3,1)      (3,2)        (3,3)
                       (1,1,2)    (4,1)        (4,2)
                       (2,1,1)    (1,1,3)      (5,1)
                       (1,1,1,1)  (2,1,2)      (1,1,4)
                                  (3,1,1)      (2,1,3)
                                  (1,1,1,2)    (2,2,2)
                                  (2,1,1,1)    (3,1,2)
                                  (1,1,1,1,1)  (4,1,1)
                                               (1,1,1,3)
                                               (2,1,1,2)
                                               (3,1,1,1)
                                               (1,1,1,1,2)
                                               (2,1,1,1,1)
                                               (1,1,1,1,1,1)

Alois P. Heinz, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The weakly decreasing version is A069916.
The version for differences instead of quotients is A325546.
The strictly increasing version is A342493.
The unordered version is A342497, ranked by A342523.
The strict unordered version is A342516.
A000005 counts constant compositions.
A000009 counts strictly increasing (or strictly decreasing) compositions.
A000041 counts weakly increasing (or weakly decreasing) compositions.
A000929 counts partitions with all adjacent parts x >= 2y.
A001055 counts factorizations.
A002843 counts compositions with all adjacent parts x <= 2y.
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.
Cf. A003242, A008965, A048004, A059966, A067824, A167606, A253249, A274199, A318991, A318992, A342527, A342528.

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..50);  # Alois P. Heinz, Mar 25 2021

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],LessEqual@@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, 50] (* Jean-François Alcover, May 19 2021, after Alois P. Heinz *)

a(21)-a(47) from Alois P. Heinz, Mar 25 2021

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

Gus Wiseman, Mar 16 2021

nonn

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).

			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)

Alois P. Heinz, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

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.

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

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 *)

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

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

Gus Wiseman, Mar 20 2021

nonn

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).

			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

Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

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.

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

A342523 Heinz numbers of integer partitions with weakly increasing first quotients.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 73, 74, 76

Offset: 1

Gus Wiseman, Mar 23 2021

nonn

Also called log-concave-up partitions.
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).

			The prime indices of 60 are {1,1,2,3}, with first quotients (1,2,3/2), so 60 is not in the sequence.
Most small numbers are in the sequence, but the sequence of non-terms together with their prime indices begins:
   18: {1,2,2}
   30: {1,2,3}
   36: {1,1,2,2}
   50: {1,3,3}
   54: {1,2,2,2}
   60: {1,1,2,3}
   70: {1,3,4}
   72: {1,1,1,2,2}
   75: {2,3,3}
   90: {1,2,2,3}
   98: {1,4,4}
  100: {1,1,3,3}

Robert Price, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The version counting strict divisor chains is A057567.
For multiplicities (prime signature) instead of quotients we have A304678.
For differences instead of quotients we have A325360 (count: A240026).
These partitions are counted by A342523 (strict: A342516, ordered: A342492).
The strictly increasing version is A342524.
The weakly decreasing version is A342526.
A000041 counts partitions (strict: A000009).
A000929 counts partitions with 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.
A318991/A318992 rank reversed partitions with/without integer quotients.
A342086 counts strict chains of divisors with strictly increasing quotients.
Cf. A000005, A002843, A056239, A067824, A112798, A124010, A130091, A238710, A253249, A325351, A325352, A342191.

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

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

Gus Wiseman, Mar 25 2021

nonn

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).

			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.

Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

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.

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

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

A358103 Quotient of the n-th divisible pair, where pairs are ordered by Heinz number. Quotient of prime indices of A318990(n).

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 5, 1, 6, 7, 8, 3, 9, 1, 4, 10, 11, 2, 12, 13, 14, 5, 15, 16, 6, 3, 17, 1, 18, 7, 2, 19, 20, 21, 22, 8, 23, 1, 24, 9, 4, 25, 26, 27, 10, 28, 29, 30, 5, 11, 31, 3, 32, 12, 33, 34, 1, 35, 36, 13, 6, 37, 2, 14, 38, 39, 15, 40, 41, 1, 42, 7, 4, 43

Offset: 1

Gus Wiseman, Nov 02 2022

nonn

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 12th divisible pair is (2,6) so a(12) = 3.

The divisible pairs are ranked by A318990, proper A339005.
Quotient of A358104 and A358105.
A different ordering is A358106.
A000040 lists the primes.
A001222 counts prime indices, distinct A001221.
A001358 lists the semiprimes, squarefree A006881.
A003963 multiplies together prime indices.
A056239 adds up prime indices.
A358192/A358193 gives quotients of semiprime indices.
Cf. A000720, A027751, A032741, A215366, A289508, A289509, A296150, A300912, A318991, A318992.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Join@@Table[Cases[primeMS[n],{x_,y_}/;Divisible[y,x]:>y/x,{0}],{n,100}]

a(n) = A358104(n)/A358105(n).

A358104 Unreduced numerator of the n-th divisible pair, where pairs are ordered by Heinz number. Greater prime index of A318990(n).

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 5, 3, 6, 7, 8, 6, 9, 4, 8, 10, 11, 6, 12, 13, 14, 10, 15, 16, 12, 9, 17, 5, 18, 14, 8, 19, 20, 21, 22, 16, 23, 6, 24, 18, 12, 25, 26, 27, 20, 28, 29, 30, 15, 22, 31, 12, 32, 24, 33, 34, 7, 35, 36, 26, 18, 37, 10, 28, 38, 39, 30, 40, 41, 8, 42

Offset: 1

Gus Wiseman, Nov 02 2022

nonn

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 12th divisible pair is (2,6) so a(12) = 6.

The divisible pairs are ranked by A318990, proper A339005.
For all semiprimes we have A338913.
The quotient of the pair is A358103.
The denominator is A358105.
The reduced version for all semiprimes is A358192, denominator A358193.
A000040 lists the primes.
A001222 counts prime indices, distinct A001221.
A001358 lists the semiprimes, squarefree A006881.
A003963 multiplies together prime indices.
A056239 adds up prime indices.
A318991 ranks divisor-chains.
Cf. A000720, A032741, A128301, A215366, A289508, A289509, A296150, A300912, A318992, A358106.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Join@@Table[Cases[primeMS[n],{x_,y_}/;Divisible[y,x]:>y,{0}],{n,1000}]

A358103(n) = a(n)/A358105(n).

A358105 Unreduced denominator of the n-th divisible pair, where pairs are ordered by Heinz number. Lesser prime index of A318990(n).

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 4, 2, 1, 1, 3, 1, 1, 1, 2, 1, 1, 2, 3, 1, 5, 1, 2, 4, 1, 1, 1, 1, 2, 1, 6, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 4, 1, 2, 1, 1, 7, 1, 1, 2, 3, 1, 5, 2, 1, 1, 2, 1, 1, 8, 1, 3, 4, 1, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Nov 02 2022

nonn

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 12th divisible pair is (2,6) so a(12) = 2.

The divisible pairs are ranked by A318990, proper A339005.
For all semiprimes we have A338912, greater A338913.
The quotient of the pair is A358103.
The reduced version for all semiprimes is A358193, numerator A358192.
A000040 lists the primes.
A001222 counts prime indices, distinct A001221.
A001358 lists the semiprimes, squarefree A006881.
A003963 multiplies together prime indices.
A056239 adds up prime indices.
A318991 ranks divisor-chains.
Cf. A000720, A027751, A128301, A215366, A289508, A289509, A296150, A300912, A318992, A358106.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Join@@Table[Cases[primeMS[n],{x_,y_}/;Divisible[y,x]:>x,{0}],{n,1000}]

A358103(n) = A358104(n)/a(n).

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

Gus Wiseman, Mar 17 2021

nonn

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).

			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)

Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

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.

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

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A342492 Number of compositions of n with weakly increasing first quotients.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A342523 Heinz numbers of integer partitions with weakly increasing first quotients.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A358103 Quotient of the n-th divisible pair, where pairs are ordered by Heinz number. Quotient of prime indices of A318990(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A358104 Unreduced numerator of the n-th divisible pair, where pairs are ordered by Heinz number. Greater prime index of A318990(n).

Views

Author

Keywords