A238710 -id:A238710 - cp's OEIS frontend

A320347 Number of partitions of n into distinct parts (a_1, a_2, ... , a_m) (a_1 > a_2 > ... > a_m and Sum_{k=1..m} a_k = n) such that a1 - a2, a2 - a3, ..., a_{m-1} - a_m are different.

1, 1, 2, 2, 3, 3, 5, 6, 6, 9, 11, 10, 15, 18, 19, 24, 31, 29, 40, 44, 51, 56, 72, 69, 90, 97, 114, 125, 154, 151, 192, 207, 237, 255, 304, 314, 377, 401, 457, 493, 573, 596, 698, 750, 845, 905, 1034, 1104, 1255, 1354, 1507, 1624, 1817, 1955, 2178, 2357, 2605, 2794, 3077, 3380

Offset: 1

Seiichi Manyama, Oct 11 2018

nonn

In other words, a(n) is the number of strict integer partitions of n with distinct first differences. - Gus Wiseman, Mar 25 2021

			n = 9
[9]        ooooooooo
------------------------------------
[8, 1]      *******o  a_1 - a_2 = 7.
            oooooooo
------------------------------------
[7, 2]       *****oo  a_1 - a_2 = 5.
             ooooooo
------------------------------------
[6, 3]        ***ooo  a_1 - a_2 = 3.
              oooooo
------------------------------------
[6, 2, 1]         *o  a_2 - a_3 = 1.
              ****oo  a_1 - a_2 = 4.
              oooooo
------------------------------------
[5, 4]         *oooo  a_1 - a_2 = 1.
               ooooo
------------------------------------
a(9) = 6.

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..500 (terms 1..100 from Seiichi Manyama)
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A000009, A320348.
The equal instead of distinct version is A049980.
The non-strict version is A325325 (ranking: A325368).
The non-strict ordered version is A325545.
The version for first quotients is A342520 (non-strict: A342514).
Cf. A003114, A003242, A005117, A238710, A342530.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&UnsameQ@@Differences[#]&]],{n,0,30}] (* Gus Wiseman, Mar 27 2021 *)

A365921 Triangle read by rows where T(n,k) is the number of integer partitions y of n such that k is the greatest member of {0..n} that is not the sum of any nonempty submultiset of y.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 0, 1, 2, 0, 4, 0, 0, 1, 2, 0, 5, 0, 0, 1, 1, 4, 0, 8, 0, 0, 0, 1, 2, 4, 0, 10, 0, 0, 0, 2, 1, 2, 7, 0, 16, 0, 0, 0, 0, 2, 1, 3, 8, 0, 20, 0, 0, 0, 0, 2, 2, 2, 4, 12, 0, 31, 0, 0, 0, 0, 0, 2, 2, 2, 5, 14, 0

Offset: 0

Gus Wiseman, Sep 30 2023

			The partition (6,2,1,1) has subset-sums 0, 1, 2, 3, 4, 6, 7, 8, 9, 10 so is counted under T(10,5).
Triangle begins:
   1
   1  0
   1  1  0
   2  0  1  0
   2  0  1  2  0
   4  0  0  1  2  0
   5  0  0  1  1  4  0
   8  0  0  0  1  2  4  0
  10  0  0  0  2  1  2  7  0
  16  0  0  0  0  2  1  3  8  0
  20  0  0  0  0  2  2  2  4 12  0
  31  0  0  0  0  0  2  2  2  5 14  0
  39  0  0  0  0  0  4  2  2  3  6 21  0
  55  0  0  0  0  0  0  4  2  4  3  9 24  0
  71  0  0  0  0  0  0  5  4  2  4  5 10 34  0
Row n = 8 counts the following partitions:
  (4211)      .  .  .  (521)   (611)  (71)   (8)     .
  (41111)              (5111)         (431)  (62)
  (3311)                                     (53)
  (3221)                                     (44)
  (32111)                                    (422)
  (311111)                                   (332)
  (22211)                                    (2222)
  (221111)
  (2111111)
  (11111111)

Steven R. Finch, Monoids of natural numbers, March 17, 2009.

Row sums are A000041.
Diagonal k = n-1 is A002865.
Column k = 1 is A126796 (complete partitions), ranks A325781.
Central diagonal n = 2k is A126796 also.
For parts instead of sums we have A339737, rank stat A339662, min A257993.
This is the triangle for the rank statistic A365920.
Latter row sums are A365924 (incomplete partitions), ranks A365830.
Column sums are A366127.
A055932 lists numbers whose prime indices cover an initial interval.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gap-free prime indices.
A238709/A238710 count partitions by least/greatest difference.
A342050/A342051 have prime indices with odd/even least gap.
A366128 gives the least non-subset-sum of prime indices.
Cf. A001522, A079068, A098743, A264401, A286469 or A286470, A339886.

nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
Table[Length[Select[IntegerPartitions[n],Max@@Prepend[nmz[#],0]==k&]],{n,0,10},{k,0,n}]

A238709 Triangular array: t(n,k) = number of partitions p = {x(1) >= x(2) >= ... >= x(k)} such that min(x(j) - x(j-1)) = k.

Original entry on oeis.org

1, 1, 1, 3, 0, 1, 4, 1, 0, 1, 7, 1, 1, 0, 1, 10, 2, 0, 1, 0, 1, 16, 2, 1, 0, 1, 0, 1, 22, 3, 1, 1, 0, 1, 0, 1, 32, 4, 2, 0, 1, 0, 1, 0, 1, 44, 5, 2, 1, 0, 1, 0, 1, 0, 1, 62, 6, 3, 1, 1, 0, 1, 0, 1, 0, 1, 83, 8, 3, 2, 0, 1, 0, 1, 0, 1, 0, 1, 113, 10, 4, 2, 1

Offset: 1

Clark Kimberling, Mar 03 2014

The first two columns are essentially A047967 and A238708. Counting the top row as row 2, the sum of numbers in row n is A000041(n) - 1.

			row 2:  1
row 3:  1 ... 1
row 4:  3 ... 0 ... 1
row 5:  4 ... 1 ... 0 ... 1
row 6:  7 ... 1 ... 1 ... 0 ... 1
row 7:  10 .. 2 ... 0 ... 1 ... 0 ... 1
row 8:  16 .. 2 ... 1 ... 0 ... 1 ... 0 ... 1
row 9:  22 .. 3 ... 1 ... 1 ... 0 ... 1 ... 0 ... 1
Let m = min(x(j) - x(j-1)); then for row 5, the 4 partitions with m = 0 are 311, 221, 2111, 11111; the 1 partition with m = 1 is 32, and the 1 partition with m = 3 is 41.

Clark Kimberling, Table of n, a(n) for n = 1..400

Cf. A238710, A238708.

z = 25; p[n_, k_] := p[n, k] = IntegerPartitions[n][[k]]; m[n_, k_] := m[n, k] = Min[-Differences[p[n, k]]]; c[n_] := Table[m[n, h], {h, 1, PartitionsP[n]}]; v = Table[Count[c[n], h], {n, 2, z}, {h, 0, n - 2}]; Flatten[v]
TableForm[v]

A355524 Minimal difference between adjacent prime indices of n > 1, or 0 if n is prime.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 2, 4, 0, 0, 0, 5, 0, 0, 0, 1, 0, 0, 3, 6, 1, 0, 0, 7, 4, 0, 0, 1, 0, 0, 0, 8, 0, 0, 0, 0, 5, 0, 0, 0, 2, 0, 6, 9, 0, 0, 0, 10, 0, 0, 3, 1, 0, 0, 7, 1, 0, 0, 0, 11, 0, 0, 1, 1, 0, 0, 0, 12, 0, 0, 4, 13, 8

Offset: 2

Gus Wiseman, Jul 10 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 9842 are {1,4,8,12}, with differences (3,4,4), so a(9842) = 3.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs found in the link are not repeated here.
Positions of first appearances are A077017 w/o the first term.
Positions of terms > 0 are A120944.
Positions of zeros are A130091.
Triangle A238353 counts m such that A056239(m) = n and a(m) = k.
For maximal difference we have A286470 or A355526.
Positions of terms > 1 are A325161.
If singletons (k) have minimal difference k we get A355525.
Positions of 1's are A355527.
Prepending 0 to the prime indices gives A355528.
A115720 and A115994 count partitions by their Durfee square.
A287352, A355533, A355534, A355536 list the differences of prime indices.
Cf. A000005, A066312, A238354, A238710, A286469, A351294, A352822, A355530, A355531, A355532.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[PrimeQ[n],0,Min@@Differences[primeMS[n]]],{n,2,100}]

A339662 Greatest gap in the partition with Heinz number n.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 0, 1, 2, 4, 0, 5, 3, 1, 0, 6, 0, 7, 2, 3, 4, 8, 0, 2, 5, 1, 3, 9, 0, 10, 0, 4, 6, 2, 0, 11, 7, 5, 2, 12, 3, 13, 4, 1, 8, 14, 0, 3, 2, 6, 5, 15, 0, 4, 3, 7, 9, 16, 0, 17, 10, 3, 0, 5, 4, 18, 6, 8, 2, 19, 0, 20, 11, 1, 7, 3, 5, 21, 2, 1, 12

Offset: 1

Gus Wiseman, Apr 20 2021

nonn

We define the greatest gap of a partition to be the greatest nonnegative integer less than the greatest part and not in the partition.
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.
Also the index of the greatest prime, up to the greatest prime index of n, not dividing n. A prime index of n is a number m such that prime(m) divides n.

George E. Andrews and David Newman, Partitions and the Minimal Excludant, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
FindStat, Dyson's crank of a partition.
Brian Hopkins, James A. Sellers, and Dennis Stanton, Dyson's Crank and the Mex of Integer Partitions, arXiv:2009.10873 [math.CO], 2020.
Wikipedia, Mex (mathematics)

Positions of first appearances are A000040.
Positions of 0's are A055932.
The version for positions of 1's in reversed binary expansion is A063250.
The prime itself (not just the index) is A079068.
The version for crank is A257989.
The minimal instead of maximal version is A257993.
The version for greatest difference is A286469 or A286470.
Positive integers by Heinz weight and image are counted by A339737.
Positions of 1's are A339886.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gap-free prime indices.
A238709/A238710 count partitions by least/greatest difference.
A342050/A342051 have prime indices with odd/even least gap.
Cf. A001223, A001522, A005117, A018818, A029707, A064391, A098743, A264401, A325351, A333214, A342192.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
maxgap[q_]:=Max@@Complement[Range[0,If[q=={},0,Max[q]]],q];
Table[maxgap[primeMS[n]],{n,100}]

a(n) = A000720(A079068(n)).

A355526 Maximal difference between adjacent prime indices of n, or k if n is the k-th prime.

Original entry on oeis.org

1, 2, 0, 3, 1, 4, 0, 0, 2, 5, 1, 6, 3, 1, 0, 7, 1, 8, 2, 2, 4, 9, 1, 0, 5, 0, 3, 10, 1, 11, 0, 3, 6, 1, 1, 12, 7, 4, 2, 13, 2, 14, 4, 1, 8, 15, 1, 0, 2, 5, 5, 16, 1, 2, 3, 6, 9, 17, 1, 18, 10, 2, 0, 3, 3, 19, 6, 7, 2, 20, 1, 21, 11, 1, 7, 1, 4, 22, 2, 0, 12

Offset: 2

Gus Wiseman, Jul 10 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 9842 are {1,4,8,12}, with differences (3,4,4), so a(9842) = 4.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs found in the link are not repeated here.
Positions of first appearances are 4 followed by A000040.
Positions of 0's are A025475, minimal version A013929.
Positions of 1's are 2 followed by A066312, minimal version A355527.
Triangle A238710 counts m such that A056239(m) = n and a(m) = k.
Prepending 0 to the prime indices gives A286469, minimal version A355528.
See also A286470, minimal version A355524.
The minimal version is A355525, triangle A238709.
The augmented version is A355532.
A001522 counts partitions with a fixed point (unproved), ranked by A352827.
A287352, A355533, A355534, A355536 list the differences of prime indices.
Cf. A000005, A047966, A091602, A238353, A279945, A325160, A325161, A352822, A351294, A355530.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[PrimeQ[n],PrimePi[n],Max@@Differences[primeMS[n]]],{n,2,100}]

A365920 Greatest non-subset-sum of the prime indices of n, or 0 if there is none.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 0, 3, 2, 4, 0, 5, 3, 4, 0, 6, 0, 7, 0, 5, 4, 8, 0, 5, 5, 5, 3, 9, 0, 10, 0, 6, 6, 6, 0, 11, 7, 7, 0, 12, 0, 13, 4, 6, 8, 14, 0, 7, 5, 8, 5, 15, 0, 7, 0, 9, 9, 16, 0, 17, 10, 7, 0, 8, 4, 18, 6, 10, 6, 19, 0, 20, 11, 7, 7, 8, 5, 21, 0, 7, 12

Offset: 1

Gus Wiseman, Sep 30 2023

nonn

This is the greatest element of {0,...,A056239(n)} that is not equal to A056239(d) for any divisor d|n, d>1. This definition is analogous to the Frobenius number of a numerical semigroup (see link), but it looks only at submultisets of a finite multiset, not all multisets of elements of a set.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 156 are {1,1,2,6}, with subset-sums 0, 1, 2, 3, 4, 6, 7, 8, 9, 10, so a(156) = 5.

Steven R. Finch, Monoids of natural numbers, March 17, 2009.

For binary indices instead of sums we have A063250.
Positions of first appearances > 2 are A065091.
Zeros are A325781, nonzeros A325798.
For prime indices instead of sums we have A339662, minimum A257993.
For least instead of greatest non-subset-sum we have A366128.
A055932 lists numbers whose prime indices cover an initial interval.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gap-free prime indices.
A238709/A238710 count partitions by least/greatest difference.
A342050/A342051 have prime indices with odd/even least gap.
Cf. A001223, A001522, A005117, A079068, A098743, A264401, A286469 or A286470, A339737, A339886.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
Table[Max@@Prepend[nmz[prix[n]],0],{n,100}]

A339737 Triangle read by rows where T(n,k) is the number of integer partitions of n with greatest gap k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 1, 1, 1, 0, 3, 1, 1, 1, 1, 0, 4, 1, 2, 2, 1, 1, 0, 5, 1, 3, 2, 2, 1, 1, 0, 6, 2, 3, 4, 3, 2, 1, 1, 0, 8, 2, 4, 5, 4, 3, 2, 1, 1, 0, 10, 2, 5, 7, 6, 5, 3, 2, 1, 1, 0, 12, 3, 6, 8, 9, 6, 5, 3, 2, 1, 1, 0, 15, 3, 8, 11, 11, 10, 7, 5, 3, 2, 1, 1, 0

Offset: 0

Gus Wiseman, Apr 20 2021

We define the greatest gap of a partition to be the greatest nonnegative integer less than the greatest part and not in the partition.

			Triangle begins:
   1
   1   0
   1   1   0
   2   0   1   0
   2   1   1   1   0
   3   1   1   1   1   0
   4   1   2   2   1   1   0
   5   1   3   2   2   1   1   0
   6   2   3   4   3   2   1   1   0
   8   2   4   5   4   3   2   1   1   0
  10   2   5   7   6   5   3   2   1   1   0
  12   3   6   8   9   6   5   3   2   1   1   0
  15   3   8  11  11  10   7   5   3   2   1   1   0
  18   4   9  13  15  13  10   7   5   3   2   1   1   0
  22   5  10  17  19  18  14  11   7   5   3   2   1   1   0
  27   5  13  20  24  23  20  14  11   7   5   3   2   1   1   0
For example, row n = 9 counts the following partitions:
  (3321)       (432)   (333)      (54)      (522)    (63)    (72)   (81)  (9)
  (22221)      (3222)  (4311)     (441)     (531)    (621)   (711)
  (32211)              (33111)    (4221)    (5211)   (6111)
  (222111)             (3111111)  (42111)   (51111)
  (321111)                        (411111)
  (2211111)
  (21111111)
  (111111111)

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
George E. Andrews and David Newman, Partitions and the Minimal Excludant, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
Brian Hopkins, James A. Sellers, and Dennis Stanton, Dyson's Crank and the Mex of Integer Partitions, arXiv:2009.10873 [math.CO], 2020.
Wikipedia, Mex (mathematics)

Column k = 0 is A000009.
Row sums are A000041.
Central diagonal is A000041.
Column k = 1 is A087897.
The version for least gap is A264401, with Heinz number encoding A257993.
The version for greatest difference is A286469 or A286470.
An encoding (of greatest gap) using Heinz numbers is A339662.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A048004 counts compositions by greatest part.
A056239 adds up prime indices, row sums of A112798.
A064391 is the version for crank.
A064428 counts partitions of nonnegative crank.
A073491 list numbers with gap-free prime indices.
A107428 counts gap-free compositions.
A238709/A238710 counts partitions by least/greatest difference.
A342050/A342051 have prime indices with odd/even least gap.
Cf. A001223, A002110, A018818, A063250, A088860, A098743, A279945.

maxgap[q_]:=Max@@Complement[Range[0,If[q=={},0,Max[q]]],q];
Table[Length[Select[IntegerPartitions[n],maxgap[#]==k&]],{n,0,15},{k,0,n}]

S(n,k)={if(k>n, O(x*x^n), x^k*(S(n-k,k+1) + 1)/(1 - x^k))}
ColGf(k,n) = {(k==0) + S(n,k+1)/prod(j=1, k-1, 1 - x^j + O(x^max(1,n-k)))}
A(n,m=n)={Mat(vector(m+1, k, Col(ColGf(k-1,n), -(n+1))))}
{ my(M=A(10)); for(i=1, #M, print(M[i,1..i])) } \\ Andrew Howroyd, Jan 13 2024

Offset corrected by Andrew Howroyd, Jan 13 2024

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]&]

cp's OEIS Frontend

A320347 Number of partitions of n into distinct parts (a_1, a_2, ... , a_m) (a_1 > a_2 > ... > a_m and Sum_{k=1..m} a_k = n) such that a1 - a2, a2 - a3, ..., a_{m-1} - a_m are different.

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A365921 Triangle read by rows where T(n,k) is the number of integer partitions y of n such that k is the greatest member of {0..n} that is not the sum of any nonempty submultiset of y.

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A238709 Triangular array: t(n,k) = number of partitions p = {x(1) >= x(2) >= ... >= x(k)} such that min(x(j) - x(j-1)) = k.

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A355524 Minimal difference between adjacent prime indices of n > 1, or 0 if n is prime.

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A339662 Greatest gap in the partition with Heinz number n.

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A355526 Maximal difference between adjacent prime indices of n, or k if n is the k-th prime.

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A365920 Greatest non-subset-sum of the prime indices of n, or 0 if there is none.

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A339737 Triangle read by rows where T(n,k) is the number of integer partitions of n with greatest gap k.

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