A000929 -id:A000929 - cp's OEIS frontend

A350842 Number of integer partitions of n with no difference -2.

1, 1, 2, 3, 4, 6, 9, 12, 16, 24, 30, 40, 54, 69, 89, 118, 146, 187, 239, 297, 372, 468, 575, 711, 880, 1075, 1314, 1610, 1947, 2359, 2864, 3438, 4135, 4973, 5936, 7090, 8466, 10044, 11922, 14144, 16698, 19704, 23249, 27306, 32071, 37639, 44019, 51457, 60113

Offset: 0

Gus Wiseman, Jan 20 2022

nonn

			The a(1) = 1 through a(7) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (211)   (41)     (51)      (52)
                    (1111)  (221)    (222)     (61)
                            (2111)   (321)     (322)
                            (11111)  (411)     (511)
                                     (2211)    (2221)
                                     (21111)   (3211)
                                     (111111)  (4111)
                                               (22111)
                                               (211111)
                                               (1111111)

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

Heinz number rankings are in parentheses below.
The version for no difference 0 is A000009.
The version for subsets of prescribed maximum is A005314.
The version for all differences < -2 is A025157, non-strict A116932.
The version for all differences > -2 is A034296, strict A001227.
The opposite version is A072670.
The version for no difference -1 is A116931 (A319630), strict A003114.
The multiplicative version is A350837 (A350838), strict A350840.
The strict case is A350844.
The complement for quotients is counted by A350846 (A350845).
A000041 = integer partitions.
A027187 = partitions of even length.
A027193 = partitions of odd length (A026424).
A323092 = double-free partitions (A320340), strict A120641.
A325534 = separable partitions (A335433).
A325535 = inseparable partitions (A335448).
A350839 = partitions with a gap and conjugate gap (A350841).
Cf. A000070, A000929, A001511, A003242, A007359, A018819, A040039, A045690, A045691, A101417, A154402, A323093.

Table[Length[Select[IntegerPartitions[n],FreeQ[Differences[#],-2]&]],{n,0,30}]

A342191 Numbers with no adjacent prime indices having quotient < 1/2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 27, 29, 30, 31, 32, 35, 36, 37, 41, 42, 43, 45, 47, 48, 49, 53, 54, 55, 59, 60, 61, 63, 64, 65, 67, 71, 72, 73, 75, 77, 79, 81, 83, 84, 89, 90, 91, 96, 97, 101, 103, 105, 107, 108, 109

Offset: 1

Gus Wiseman, Mar 05 2021

nonn

Also Heinz numbers of integer partitions with no adjacent parts having quotient > 2 (counted by A342094). The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
      1: {}            18: {1,2,2}         42: {1,2,4}
      2: {1}           19: {8}             43: {14}
      3: {2}           21: {2,4}           45: {2,2,3}
      4: {1,1}         23: {9}             47: {15}
      5: {3}           24: {1,1,1,2}       48: {1,1,1,1,2}
      6: {1,2}         25: {3,3}           49: {4,4}
      7: {4}           27: {2,2,2}         53: {16}
      8: {1,1,1}       29: {10}            54: {1,2,2,2}
      9: {2,2}         30: {1,2,3}         55: {3,5}
     11: {5}           31: {11}            59: {17}
     12: {1,1,2}       32: {1,1,1,1,1}     60: {1,1,2,3}
     13: {6}           35: {3,4}           61: {18}
     15: {2,3}         36: {1,1,2,2}       63: {2,2,4}
     16: {1,1,1,1}     37: {12}            64: {1,1,1,1,1,1}
     17: {7}           41: {13}            65: {3,6}

The multiplicative version (squared instead of doubled) for prime factors is A253784.
These are the Heinz numbers of the partitions counted by A342094.
A003114 counts partitions with adjacent parts differing by more than 1.
A034296 counts partitions with adjacent parts differing by at most 1.
A038548 counts inferior or superior divisors, listed by A161906 or A161908.
Cf. A000929, A003242, A056239, A056924, A112798, A154402, A167606, A337135, A342085, A342096, A342098.

Select[Range[100],Min[Divide@@@Partition[PrimePi/@First/@FactorInteger[#],2,1]]>=1/2&]

A342083 Number of chains of strictly inferior divisors from n to 1.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 28 2021

nonn

We define a divisor d|n to be strictly inferior if d < n/d. Strictly inferior divisors are counted by A056924 and listed by A341674.
These chains have first-quotients (in analogy with first-differences) that are term-wise > their decapitation (maximum element removed). Equivalently, x > y^2 for all adjacent x, y. For example, the divisor chain q = 60/6/2/1 has first-quotients (10,3,2), which are > (6,2,1), so q is counted under a(60).
Also the number of factorizations of n where each factor is greater than the product of all previous factors.

			The a(n) chains for n = 2, 6, 12, 24, 42, 48, 60, 72:
  2/1  6/1    12/1    24/1    42/1      48/1      60/1      72/1
       6/2/1  12/2/1  24/2/1  42/2/1    48/2/1    60/2/1    72/2/1
              12/3/1  24/3/1  42/3/1    48/3/1    60/3/1    72/3/1
                      24/4/1  42/6/1    48/4/1    60/4/1    72/4/1
                              42/6/2/1  48/6/1    60/5/1    72/6/1
                                        48/6/2/1  60/6/1    72/8/1
                                                  60/6/2/1  72/6/2/1
                                                            72/8/2/1
The a(n) factorizations for n = 2, 6, 12, 24, 42, 48, 60, 72:
  2  6    12   24    42     48     60      72
     2*3  2*6  3*8   6*7    6*8    2*30    8*9
          3*4  4*6   2*21   2*24   3*20    2*36
               2*12  3*14   3*16   4*15    3*24
                     2*3*7  4*12   5*12    4*18
                            2*3*8  6*10    6*12
                                   2*3*10  2*4*9
                                           2*3*12

Amiram Eldar, Table of n, a(n) for n = 1..10000

The restriction to powers of 2 is A040039.
Not requiring strict inferiority gives A074206 (ordered factorizations).
The weakly inferior version is A337135.
The strictly superior version is A342084.
The weakly superior version is A342085.
The additive version is A342098, or A000929 allowing equality.
A000005 counts divisors.
A001055 counts factorizations.
A003238 counts chains of divisors summing to n-1, with strict case A122651.
A038548 counts inferior (or superior) divisors.
A056924 counts strictly inferior (or strictly superior) divisors.
A067824 counts strict chains of divisors starting with n.
A167865 counts strict chains of divisors > 1 summing to n.
A207375 lists central divisors.
A253249 counts strict chains of divisors.
A334996 counts ordered factorizations by product and length.
A334997 counts chains of divisors of n by length.
A342086 counts chains of divisors with strictly increasing quotients > 1.
- Inferior: A033676, A066839, A072499, A161906.
- Superior: A033677, A070038, A161908.
- Strictly Inferior: A060775, A070039, A333806, A341674.
- Strictly Superior: A048098, A064052, A140271, A238535, A341673.
Cf. A000203, A001248, A002033, A006530, A018819, A020639, A045690, A337105, A342087, A342094, A342095, A342096, A342097.

cen[n_]:=If[n==1,{{1}},Prepend[#,n]&/@Join@@cen/@Select[Divisors[n],#

G.f.: x + Sum_{k>=1} a(k) * x^(k*(k + 1)) / (1 - x^k). - Ilya Gutkovskiy, Nov 03 2021

A342084 Number of chains of distinct strictly superior divisors starting with n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 9, 1, 2, 2, 4, 1, 7, 1, 6, 2, 2, 2, 10, 1, 2, 2, 9, 1, 6, 1, 4, 4, 2, 1, 19, 1, 4, 2, 4, 1, 8, 2, 9, 2, 2, 1, 20, 1, 2, 4, 10, 2, 6, 1, 4, 2, 7, 1, 29, 1, 2, 4, 4, 2, 6, 1, 19, 3, 2, 1, 19, 2

Offset: 1

Gus Wiseman, Feb 28 2021

nonn

We define a divisor d|n to be strictly superior if d > n/d. Strictly superior divisors are counted by A056924 and listed by A341673.
These chains have first-quotients (in analogy with first-differences) that are term-wise < their decapitation (maximum element removed). Equivalently, x < y^2 for all adjacent x, y. For example, the divisor chain q = 30/6/3 has first-quotients (5,2), which are < (6,3), so q is counted under a(30).
Also the number of ordered factorizations of n where each factor is less than the product of all previous factors.

			The a(n) chains for n = 2, 6, 12, 16, 24, 30, 32, 36:
  2  6    12      16      24         30       32         36
     6/3  12/4    16/8    24/6       30/6     32/8       36/9
          12/6    16/8/4  24/8       30/10    32/16      36/12
          12/6/3          24/12      30/15    32/8/4     36/18
                          24/6/3     30/6/3   32/16/8    36/12/4
                          24/8/4     30/10/5  32/16/8/4  36/12/6
                          24/12/4    30/15/5             36/18/6
                          24/12/6                        36/18/9
                          24/12/6/3                      36/12/6/3
                                                         36/18/6/3
The a(n) ordered factorizations for n = 2, 6, 12, 16, 24, 30, 32, 36:
  2  6    12     16     24       30     32       36
     3*2  4*3    8*2    6*4      6*5    8*4      9*4
          6*2    4*2*2  8*3      10*3   16*2     12*3
          3*2*2         12*2     15*2   4*2*4    18*2
                        3*2*4    3*2*5  8*2*2    4*3*3
                        4*2*3    5*2*3  4*2*2*2  6*2*3
                        4*3*2    5*3*2           6*3*2
                        6*2*2                    9*2*2
                        3*2*2*2                  3*2*2*3
                                                 3*2*3*2

Amiram Eldar, Table of n, a(n) for n = 1..10000

The restriction to powers of 2 is A045690, with reciprocal version A040039.
The inferior version is A337135.
The strictly inferior version is A342083.
The superior version is A342085.
The additive version allowing equality is A342094 or A342095.
The additive version is A342096 or A342097.
A000005 counts divisors.
A001055 counts factorizations.
A003238 counts divisibility chains summing to n-1, with strict case A122651.
A038548 counts inferior (or superior) divisors.
A056924 counts strictly inferior (or strictly superior) divisors.
A067824 counts strict chains of divisors starting with n.
A074206 counts strict chains of divisors from n to 1 (also ordered factorizations).
A167865 counts strict chains of divisors > 1 summing to n.
A207375 lists central divisors.
A253249 counts strict chains of divisors.
A334996 counts ordered factorizations by product and length.
A334997 counts chains of divisors of n by length.
- Inferior: A033676, A063962, A066839, A072499, A161906.
- Superior: A033677, A070038, A161908, A341591.
- Strictly Inferior: A060775, A070039, A333806, A341674.
- Strictly Superior: A064052/A048098, A140271, A238535, A341642, A341673.
Cf. A000203, A000929, A001248, A006530, A018819, A020639, A169594, A337105, A342098.

ceo[n_]:=Prepend[Prepend[#,n]&/@Join@@ceo/@Select[Most[Divisors[n]],#>n/#&],{n}];
Table[Length[ceo[n]],{n,100}]

a(2^n) = A045690(n).

A274199 Limiting reverse row of the array A274190.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 12, 19, 29, 44, 67, 101, 152, 228, 342, 511, 763, 1138, 1695, 2523, 3752, 5578, 8287, 12307, 18272, 27119, 40241, 59700, 88556, 131340, 194772, 288815, 428229, 634900, 941263, 1395397, 2068560, 3066372, 4545387, 6737633, 9987026, 14803303

Offset: 0

Clark Kimberling, Jun 13 2016

The triangular array (g(n,k)) at A274190 is defined as follows:  g(n,k) = 1 for n >= 0; g(n,k) = 0 if k > n; g(n,k) = g(n-1,k-1) + g(n-1,2k) for n > 0, k > 1.
From Gus Wiseman, Mar 12 2021: (Start)
Also (apparently) the number of compositions of n where all adjacent parts (x, y), satisfy x < 2y. For example, the a(1) = 1 through a(6) = 12 compositions are:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (12)   (13)    (14)     (15)
             (111)  (22)    (23)     (24)
                    (112)   (32)     (33)
                    (1111)  (113)    (114)
                            (122)    (123)
                            (1112)   (132)
                            (11111)  (222)
                                     (1113)
                                     (1122)
                                     (11112)
                                     (111111)
(End)

			Row (g(14,k)):  1, 51, 73, 69, 55, 40, 28, 19, 12, 8, 5, 3, 2, 1, 1; the reversal is 1 1 2 3 5 8 12 19 28 ..., which agrees with A274199 up to 19.

Daniel Gabric and Jeffrey Shallit, Smallest and Largest Block Palindrome Factorizations, arXiv:2302.13147 [math.CO], 2023.

Cf. A274190, A274200, A274201.

Cf. A000929, A003242, A154402, A224957, A342094, A342095, A342096, A342097, A342098, A342191, A342330-A342342.

g[n_, 0] = g[n, 0] = 1;
g[n_, k_] := g[n, k] = If[k > n, 0, g[n - 1, k - 1] + g[n - 1, 2 k]];
z = 300; u = Reverse[Table[g[z, k], {k, 0, z}]];
z = 301; v = Reverse[Table[g[z, k], {k, 0, z}]];
w = Join[{1}, Intersection[u, v]] (* A274199 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],And@@Table[#[[i]]<2*#[[i-1]],{i,2,Length[#]}]&]],{n,15}] (* Gus Wiseman, Mar 12 2021 *)

A342337 Number of integer partitions of n with all adjacent parts (x, y) satisfying either x = y or x = 2y.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 7, 6, 9, 10, 12, 11, 19, 14, 20, 24, 27, 24, 37, 31, 44, 45, 49, 48, 71, 61, 72, 80, 92, 84, 118, 102, 128, 132, 144, 151, 191, 166, 197, 211, 244, 226, 287, 263, 313, 330, 348, 347, 435, 399, 462, 476, 524, 508, 614, 591, 674, 680, 732, 731, 890, 814, 916, 966, 1042, 1032, 1188, 1135, 1280, 1303

Offset: 0

Gus Wiseman, Mar 10 2021

nonn

			The a(1) = 1 through a(9) = 10 partitions:
  1   2    3     4      5       6        7         8          9
      11   21    22     221     33       421       44         63
           111   211    2111    42       2221      422        333
                 1111   11111   222      22111     2222       4221
                                2211     211111    4211       22221
                                21111    1111111   22211      42111
                                111111             221111     222111
                                                   2111111    2211111
                                                   11111111   21111111
                                                              111111111

Alois P. Heinz, Table of n, a(n) for n = 0..10000

The first condition alone gives A000005 (for partitions).
The second condition alone gives A154402 (for partitions).
The Heinz numbers of these partitions are given by A342339.
A000929 counts partitions with adjacent parts x >= 2y.
A002843 counts compositions with adjacent parts x <= 2y.
A224957 counts compositions with x <= 2y and y <= 2x (strict: A342342).
A274199 counts compositions with adjacent parts x < 2y.
A342094 counts partitions with adjacent parts x <= 2y (strict: A342095).
A342096 counts partitions without adjacent x >= 2y (strict: A342097).
A342098 counts partitions with adjacent parts x > 2y.
A342330 counts compositions with x < 2y and y < 2x (strict: A342341).
A342331 counts compositions with adjacent parts x = 2y or y = 2x.
A342332 counts compositions with adjacent parts x > 2y or y > 2x.
A342333 counts compositions with adjacent parts x >= 2y or y >= 2x.
A342335 counts compositions with adjacent parts x >= 2y or y = 2x.
A342338 counts compositions with adjacent parts x < 2y and y <= 2x.
Cf. A003114, A003242, A034296, A167606, A342083, A342084, A342087, A342191, A342334, A342336, A342340.

b:= proc(n, i) option remember; `if`(n=0, 1, add(b(n-j, j),
      j=`if`(i=0, 1..n, select(x-> x<=n, [i, 2*i]))))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, May 24 2021

Table[Length[Select[IntegerPartitions[n],And@@Table[#[[i]]==#[[i-1]]||#[[i-1]]==2*#[[i]],{i,2,Length[#]}]&]],{n,0,30}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n - j, j],
     {j, If[i == 0, Range[n], Select[{i, 2i}, # <= n&]]}]];
a[n_] := b[n, 0];
a /@ Range[0, 80] (* Jean-François Alcover, Jun 03 2021, after Alois P. Heinz *)

A224957 Number of compositions [p(1), p(2), ..., p(k)] of n such that p(j) <= 2p(j-1) and p(j-1) <= 2p(j).

Original entry on oeis.org

1, 1, 2, 4, 6, 11, 19, 31, 54, 92, 154, 266, 454, 771, 1319, 2249, 3834, 6550, 11176, 19069, 32558, 55567, 94838, 161891, 276325, 471659, 805102, 1374234, 2345724, 4004031, 6834605, 11666260, 19913668, 33991462, 58021534, 99039592, 169055094, 288567886, 492569833, 840790082

Offset: 0

Joerg Arndt, Apr 21 2013

nonn

			There are a(6) = 19 such compositions of 6:
01:  [ 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 2 ]
03:  [ 1 1 1 2 1 ]
04:  [ 1 1 2 1 1 ]
05:  [ 1 1 2 2 ]
06:  [ 1 2 1 1 1 ]
07:  [ 1 2 1 2 ]
08:  [ 1 2 2 1 ]
09:  [ 1 2 3 ]
10:  [ 2 1 1 1 1 ]
11:  [ 2 1 1 2 ]
12:  [ 2 1 2 1 ]
13:  [ 2 2 1 1 ]
14:  [ 2 2 2 ]
15:  [ 2 4 ]
16:  [ 3 2 1 ]
17:  [ 3 3 ]
18:  [ 4 2 ]
19:  [ 6 ]

Alois P. Heinz, Table of n, a(n) for n = 0..4306

The case of strict relations is A342330, with strict case A342341.
The strict case is A342342.
A000929 counts partitions with adjacent parts x >= 2y.
A002843 counts compositions with adjacent parts x <= 2y.
A045690 counts sets with maximum n with adjacent elements y < 2x.
A154402 counts partitions with adjacent parts x = 2y.
A274199 counts compositions with adjacent parts x < 2y.
A342094 counts partitions with adjacent parts x <= 2y (strict: A342095).
A342096 counts partitions without adjacent x >= 2y (strict: A342097).
A342098 counts partitions with adjacent parts x > 2y.
A342331 counts compositions with adjacent parts x = 2y or y = 2x.
A342332 counts compositions with adjacent parts x > 2y or y > 2x.
A342333 counts compositions with adjacent parts x >= 2y or y >= 2x.
A342334 counts compositions with adjacent parts x >= 2y or y > 2x.
A342335 counts compositions with adjacent parts x >= 2y or y = 2x.
A342336 counts compositions with adjacent parts x > 2y or y = 2x.
A342337 counts partitions with adjacent parts x = y or x = 2y.
A342338 counts compositions with adjacent parts x < 2y and y <= 2x.
A342340 counts compositions with adjacent x = y or x = 2y or y = 2x.
Cf. A003114, A003242, A034296, A167606, A342191, A342339.

b:= proc(n, i) option remember; `if`(n=0, 1, add(
      b(n-j, j), j=`if`(i=0, 1..n, ceil(i/2)..min(n, 2*i))))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..42);  # Alois P. Heinz, Mar 15 2021

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],And@@Table[#[[i]]<=2*#[[i-1]]&&#[[i-1]]<=2*#[[i]],{i,2,Length[#]}]&]],{n,15}] (* Gus Wiseman, Mar 12 2021 *)
b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n - j, j], {j, If[i == 0, Range[n], Range[Ceiling[i/2], Min[n, 2*i]]]}]];
a[n_] := b[n, 0];
a /@ Range[0, 42] (* Jean-François Alcover, May 24 2021, after Alois P. Heinz *)

Name corrected by Gus Wiseman, Mar 11 2021

A337135 a(1) = 1; for n > 1, a(n) = Sum_{d|n, d <= sqrt(n)} a(d).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 4, 2, 2, 1, 5, 2, 2, 2, 4, 1, 4, 1, 4, 2, 2, 2, 7, 1, 2, 2, 5, 1, 5, 1, 4, 3, 2, 1, 7, 2, 3, 2, 4, 1, 5, 2, 5, 2, 2, 1, 8, 1, 2, 3, 6, 2, 5, 1, 4, 2, 4, 1, 9, 1, 2, 3, 4, 2, 5, 1, 7, 4, 2, 1, 8, 2, 2, 2, 6, 1, 8, 2, 4, 2, 2, 2

Offset: 1

Ilya Gutkovskiy, Nov 21 2020

nonn

From Gus Wiseman, Mar 05 2021: (Start)
This sequence counts all of the following essentially equivalent things:
1. Chains of distinct inferior divisors from n to 1, where a divisor d|n is inferior if d <= n/d. Inferior divisors are counted by A038548 and listed by A161906.
2. Chains of divisors from n to 1 whose first-quotients (in analogy with first-differences) are term-wise greater than or equal to their decapitation (maximum element removed). For example, the divisor chain q = 60/4/2/1 has first-quotients (15,2,2), which are >= (4,2,1), so q is counted under a(60).
3. Chains of divisors from n to 1 such that x >= y^2 for all adjacent x, y.
4. Factorizations of n where each factor is greater than or equal to the product of all previous factors.
(End)

			From _Gus Wiseman_, Mar 05 2021: (Start)
The a(n) chains for n = 1, 2, 4, 12, 16, 24, 36, 60:
  1  2/1  4/1    12/1    16/1      24/1      36/1      60/1
          4/2/1  12/2/1  16/2/1    24/2/1    36/2/1    60/2/1
                 12/3/1  16/4/1    24/3/1    36/3/1    60/3/1
                         16/4/2/1  24/4/1    36/4/1    60/4/1
                                   24/4/2/1  36/6/1    60/5/1
                                             36/4/2/1  60/6/1
                                             36/6/2/1  60/4/2/1
                                                       60/6/2/1
The a(n) factorizations for n = 2, 4, 12, 16, 24, 36, 60:
    2  4    12   16     24     36     60
       2*2  2*6  2*8    3*8    4*9    2*30
            3*4  4*4    4*6    6*6    3*20
                 2*2*4  2*12   2*18   4*15
                        2*2*6  3*12   5*12
                               2*2*9  6*10
                               2*3*6  2*2*15
                                      2*3*10
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..65536

Cf. A002033, A008578 (positions of 1's), A068108.
The restriction to powers of 2 is A018819.
Not requiring inferiority gives A074206 (ordered factorizations).
The strictly inferior version is A342083.
The strictly superior version is A342084.
The weakly superior version is A342085.
The additive version is A000929, or A342098 forbidding equality.
A000005 counts divisors, with sum A000203.
A001055 counts factorizations.
A003238 counts chains of divisors summing to n-1, with strict case A122651.
A038548 counts inferior (or superior) divisors.
A056924 counts strictly inferior (or strictly superior) divisors.
A067824 counts strict chains of divisors starting with n.
A167865 counts strict chains of divisors > 1 summing to n.
A207375 lists central divisors.
A253249 counts strict chains of divisors.
A334996 counts ordered factorizations by product and length.
A334997 counts chains of divisors of n by length.
A342086 counts strict factorizations of divisors.
- Inferior: A033676, A066839, A072499, A161906.
- Superior: A033677, A070038, A161908.
- Strictly Inferior: A060775, A070039, A333806, A341674.
- Strictly Superior: A048098, A064052, A140271, A238535, A341673.
Cf. A001248, A006530, A020639, A040039, A045690, A337105, A342087, A342094, A342095, A342096, A342097.

a:= proc(n) option remember; `if`(n=1, 1, add(
      `if`(d<=n/d, a(d), 0), d=numtheory[divisors](n)))
    end:
seq(a(n), n=1..128);  # Alois P. Heinz, Jun 24 2021

a[1] = 1; a[n_] := a[n] = DivisorSum[n, a[#] &, # <= Sqrt[n] &]; Table[a[n], {n, 95}]
(* second program *)
asc[n_]:=Prepend[#,n]&/@Prepend[Join@@Table[asc[d],{d,Select[Divisors[n],#Gus Wiseman, Mar 05 2021 *)

G.f.: Sum_{k>=1} a(k) * x^(k^2) / (1 - x^k).

a(2^n) = A018819(n). - Gus Wiseman, Mar 08 2021

A342087 Number of chains of divisors starting with n and having no adjacent parts x <= y^2.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 4, 2, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 8, 2, 4, 4, 6, 2, 8, 2, 6, 4, 4, 4, 8, 2, 4, 4, 8, 2, 10, 2, 6, 6, 4, 2, 12, 2, 6, 4, 6, 2, 10, 4, 8, 4, 4, 2, 14, 2, 4, 6, 6, 4, 10, 2, 6, 4, 8, 2, 16, 2, 4, 6, 6, 4, 10, 2, 12, 4, 4, 2, 14

Offset: 1

Gus Wiseman, Mar 05 2021

nonn

An alternative wording: Number of chains of divisors starting with n and having all adjacent parts x > y^2.

			The chains for n = 1, 2, 6, 12, 24, 42, 48:
   1    2      6        12        24        42          48
        2/1    6/1      12/1      24/1      42/1        48/1
               6/2      12/2      24/2      42/2        48/2
               6/2/1    12/3      24/3      42/3        48/3
                        12/2/1    24/4      42/6        48/4
                        12/3/1    24/2/1    42/2/1      48/6
                                  24/3/1    42/3/1      48/2/1
                                  24/4/1    42/6/1      48/3/1
                                            42/6/2      48/4/1
                                            42/6/2/1    48/6/1
                                                        48/6/2
                                                        48/6/2/1

Amiram Eldar, Table of n, a(n) for n = 1..10000

The restriction to powers of 2 is A018819.
Not requiring strict inferiority gives A067824.
The weakly inferior version is twice A337135.
The case ending with 1 is counted by A342083.
The strictly superior version is A342084.
The weakly superior version is A342085.
The additive version is A342098, or A000929 allowing equality.
A000005 counts divisors, with sum A000203.
A001055 counts factorizations.
A003238 counts chains of divisors summing to n-1, with strict case A122651.
A038548 counts inferior (or superior) divisors.
A056924 counts strictly inferior (or strictly superior) divisors.
A067824 counts strict chains of divisors starting with n.
A074206 counts ordered factorizations.
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.
Cf. A002033, A006530, A018819, A020639, A337105, A341674, A342086, A342094, A342095, A342097.

cem[n_]:=Prepend[Prepend[#,n]&/@Join@@cem/@Most[Divisors[n]],{n}];
Table[Length[Select[cem[n],And@@Thread[Divide@@@Partition[#,2,1]>Rest[#]]&]],{n,30}]

For n > 1, a(n) = 2*A342083(n).

A342330 Number of compositions of n with all adjacent parts (x,y) satisfying x < 2y and y < 2x.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 4, 7, 9, 11, 17, 23, 32, 44, 63, 91, 127, 180, 255, 363, 516, 732, 1044, 1485, 2109, 3002, 4277, 6089, 8660, 12323, 17550, 24986, 35562, 50628, 72084, 102616, 146077, 207980, 296114, 421555, 600153, 854469, 1216543, 1731983, 2465842, 3510713

Offset: 0

Gus Wiseman, Mar 09 2021

nonn

Each quotient of adjacent parts is between 1/2 and 2 exclusive.

			The a(1) = 1 through a(9) = 11 partitions:
  1   2    3     4      5       6        7         8          9
      11   111   22     23      33       34        35         45
                 1111   32      222      43        44         54
                        11111   111111   223       53         234
                                         232       233        333
                                         322       323        432
                                         1111111   332        2223
                                                   2222       2232
                                                   11111111   2322
                                                              3222
                                                              111111111

Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 1001 terms from Andrew Howroyd)

The version allowing equality is A224957.
The unordered version (partitions) is A342096, with strict case A342097.
Reversing operators and changing 'and' into 'or' gives A342332.
The version allowing partial equality is A342338.
The strict case is A342341.
A000929 counts partitions with all adjacent parts x >= 2y.
A002843 counts compositions with all adjacent parts x <= 2y.
A154402 counts partitions with all adjacent parts x = 2y.
A274199 counts compositions with all adjacent parts x < 2y.
A342094 counts partitions with all adjacent parts x <= 2y (strict: A342095).
A342098 counts partitions with all adjacent parts x > 2y.
A342331 counts compositions where each part is twice or half the prior.
A342335 counts compositions with all adjacent parts x >= 2y or y = 2x.
A342337 counts compositions with all adjacent parts x = y or x = 2y.
Cf. A003114, A003242, A034296, A167606, A342083, A342084, A342087, A342191, A342333, A342334, A342336, A342339, A342340.

b:= proc(n, i) option remember; `if`(n=0, 1, add(b(n-j, j)
      , j=`if`(i=0, 1..n, floor(i/2)+1..min(n, 2*i-1))))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..45);  # Alois P. Heinz, Mar 15 2021

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],And@@Table[#[[i]]<2*#[[i-1]]&&#[[i-1]]<2*#[[i]],{i,2,Length[#]}]&]],{n,0,15}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n - j, j], {j, If[i == 0, 1, Floor[i/2] + 1], If[i == 0, n, Min[n, 2i - 1]]}]];
a[n_] := b[n, 0];
a /@ Range[0, 45] (* Jean-François Alcover, May 09 2021, after Alois P. Heinz *)

C(n, pred)={my(M=matid(n)); for(k=1, n, for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); sum(q=1, n, M[q, ])}
seq(n)={concat([1], C(n, (i,j)->i<2*j && j<2*i))} \\ Andrew Howroyd, Mar 13 2021

Terms a(31) and beyond from Andrew Howroyd, Mar 13 2021

cp's OEIS Frontend

A350842 Number of integer partitions of n with no difference -2.

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A342191 Numbers with no adjacent prime indices having quotient < 1/2.

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A342083 Number of chains of strictly inferior divisors from n to 1.

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A342084 Number of chains of distinct strictly superior divisors starting with n.

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A274199 Limiting reverse row of the array A274190.

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A342337 Number of integer partitions of n with all adjacent parts (x, y) satisfying either x = y or x = 2y.

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A224957 Number of compositions [p(1), p(2), ..., p(k)] of n such that p(j) <= 2*p(j-1) and p(j-1) <= 2*p(j).

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A337135 a(1) = 1; for n > 1, a(n) = Sum_{d|n, d <= sqrt(n)} a(d).

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A224957 Number of compositions [p(1), p(2), ..., p(k)] of n such that p(j) <= 2p(j-1) and p(j-1) <= 2p(j).