A034296 -id:A034296 - cp's OEIS frontend

A239955 Number of partitions p of n such that (number of distinct parts of p) <= max(p) - min(p).

0, 0, 0, 0, 1, 2, 4, 7, 12, 17, 27, 38, 54, 75, 104, 137, 187, 245, 322, 418, 542, 691, 887, 1121, 1417, 1777, 2228, 2767, 3441, 4247, 5235, 6424, 7871, 9594, 11688, 14173, 17168, 20723, 24979, 30008, 36010, 43085, 51479, 61357, 73032, 86718, 102852, 121718

Offset: 0

Clark Kimberling, Mar 30 2014

From Gus Wiseman, Jun 26 2022: (Start)
Also the number of partitions of n with at least one gap, i.e., partitions whose parts do not form a contiguous interval. These partitions are ranked by A073492. For example, the a(0) = 0 through a(8) = 12 partitions are:
  .  .  .  .  (31)  (41)   (42)    (52)     (53)
                    (311)  (51)    (61)     (62)
                           (411)   (331)    (71)
                           (3111)  (421)    (422)
                                   (511)    (431)
                                   (4111)   (521)
                                   (31111)  (611)
                                            (3311)
                                            (4211)
                                            (5111)
                                            (41111)
                                            (311111)
Also the number of non-constant partitions of n with a repeated non-maximal part, ranked by A065201. The a(0) = 0 through a(8) = 12 partitions are:
  .  .  .  .  (211)  (311)   (411)    (322)     (422)
                     (2111)  (2211)   (511)     (611)
                             (3111)   (3211)    (3221)
                             (21111)  (4111)    (3311)
                                      (22111)   (4211)
                                      (31111)   (5111)
                                      (211111)  (22211)
                                                (32111)
                                                (41111)
                                                (221111)
                                                (311111)
                                                (2111111)
(End)

			a(6) counts these 4 partitions:  51, 42, 411, 3111.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 201 terms from John Tyler Rascoe)

Cf. A239954, A239956, A239958.
The complement is counted by A034296 (strict A137793), ranked by A073491.
These partitions are ranked by A073492, conjugate A065201.
Applying the condition to the conjugate gives A350839, ranked by A350841.
A000041 counts integer partitions, strict A000009.
A090858 counts partitions with a single hole, ranked by A325284.
A116931 counts partitions with differences != -1, strict A003114.
A116932 counts partitions with differences != -1 or -2, strict A025157.
Cf. A000070, A001227, A008284, A144300, A183558, A321440.

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(b(n-i*j, i-1), j=1..n/i)))
    end:
a:= n-> combinat[numbpart](n)-add(b(n, k), k=0..n):
seq(a(n), n=0..47);  # Alois P. Heinz, Aug 18 2025

z = 60; d[p_] := d[p] = Length[DeleteDuplicates[p]]; f[p_] := f[p] = Max[p] - Min[p]; g[n_] := g[n] = IntegerPartitions[n];
Table[Count[g[n], p_ /; d[p] < f[p]], {n, 0, z}]  (*A239954*)
Table[Count[g[n], p_ /; d[p] <= f[p]], {n, 0, z}] (*A239955*)
Table[Count[g[n], p_ /; d[p] == f[p]], {n, 0, z}] (*A239956*)
Table[Count[g[n], p_ /; d[p] > f[p]], {n, 0, z}]  (*A034296*)
Table[Count[g[n], p_ /; d[p] >= f[p]], {n, 0, z}] (*A239958*)
(* second program *)
Table[Length[Select[IntegerPartitions[n],Min@@Differences[#]<-1&]],{n,0,30}] (* Gus Wiseman, Jun 26 2022 *)

qs(a,q,n) = {prod(k=0,n,1-a*q^k)}
A_q(N) = {if(N<4, vector(N+1,i,0), my(q='q+O('q^(N-2)), g= sum(i=2,N+1, q^i/qs(q,q,i-1)*sum(j=1,i-1, q^(2*j)*qs(q^2,q^2,j-2)))); concat([0,0,0,0], Vec(g)))} \\ John Tyler Rascoe, Aug 16 2025

a(n) = A000041(n) - A034296(n).

G.f.: Sum_{i>1} q^i/(q;q){i-1} * Sum{j=1..i-1} (q^2;q^2){j-2} where (a;q)_k = Product{i>=0..k} (1-a*q^i). - John Tyler Rascoe, Aug 16 2025

A375128 Irregular triangle read by rows where row n lists the minima of maximal strictly increasing runs in the weakly increasing prime indices of n.

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 4, 1, 1, 1, 2, 2, 1, 5, 1, 1, 6, 1, 2, 1, 1, 1, 1, 7, 1, 2, 8, 1, 1, 2, 1, 9, 1, 1, 1, 3, 3, 1, 2, 2, 2, 1, 1, 10, 1, 11, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 12, 1, 2, 1, 1, 1, 13, 1, 14, 1, 1, 2, 2, 1, 15, 1, 1, 1, 1, 4, 4, 1, 3, 2, 1, 1, 16

Offset: 1

Gus Wiseman, Aug 04 2024

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 minima of strictly increasing runs in a sequence are obtained by splitting it into maximal strictly increasing subsequences and taking the first term of each.

			The prime indices of 540 are {1,1,2,2,2,3}, with strictly increasing runs ({1},{1,2},{2},{2,3}), with minima (1,1,2,2), which is row 540.
Triangle begins:
   1:
   2:  1
   3:  2
   4:  1  1
   5:  3
   6:  1
   7:  4
   8:  1  1  1
   9:  2  2
  10:  1
  11:  5
  12:  1  1
  13:  6
  14:  1
  15:  2
  16:  1  1  1  1

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Row-minima are A055396.
Row-sums are A374706.
Row-lengths are A375136.
For leaders of constant runs we have A304038, row-sums A066328.
For compositions we have A374683, row-sums of A374684 (length A124768).
A112798 lists prime indices:
- length A001222, distinct A001221
- leader A055396
- sum A056239
- reverse A296150
Cf. A034296, A141199, A218482, A279790, A320324, A333213, A358836, A374634, A374700, A374758, A375133.

Table[If[n==1,{},First/@Split[Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]],Less]],{n,100}]

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 *)

A034297 Number of ordered positive integer solutions (m_1, m_2, ..., m_k) (for some k) to Sum_{i=1..k} m_i=n with |m_i-m_{i-1}| <= 1 for i = 2 ... k.

Original entry on oeis.org

1, 1, 2, 4, 6, 11, 17, 29, 47, 78, 130, 215, 357, 595, 990, 1651, 2748, 4584, 7643, 12744, 21256, 35451, 59133, 98636, 164531, 274463, 457837, 763746, 1274060, 2125356, 3545491, 5914545, 9866602, 16459421, 27457549, 45804648, 76411272, 127469285, 212644336

Offset: 0

Erich Friedman

nonn

Compositions of n where successive parts differ by at most 1, see example. [Joerg Arndt, Dec 10 2012]

			From _Joerg Arndt_, Dec 10 2012: (Start)
The a(6) = 17 such compositions of 6 are
[ #]     composition
[ 1]    [ 1 1 1 1 1 1 ]
[ 2]    [ 1 1 1 1 2 ]
[ 3]    [ 1 1 1 2 1 ]
[ 4]    [ 1 1 2 1 1 ]
[ 5]    [ 1 1 2 2 ]
[ 6]    [ 1 2 1 1 1 ]
[ 7]    [ 1 2 1 2 ]
[ 8]    [ 1 2 2 1 ]
[ 9]    [ 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]    [ 3 2 1 ]
[16]    [ 3 3 ]
[17]    [ 6 ]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..4500
Jia Huang, Compositions with restricted parts, arXiv:1812.11010 [math.CO], 2018.

Cf. A003116, A034296.
Column k=1 of A214246, A214248.
Row sums of A309939.

b:= proc(n, i) option remember;
      `if`(n=i, 1, `if`(n<0 or i<1, 0, add(b(n-i, i+j), j=-1..1)))
    end:
a:= n-> add(b(n, k), k=0..n):
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 06 2012

b[n_, i_] := b[n, i] = If[n == i, 1, If[n<0 || i<1, 0, Sum[b[n-i, i+j], {j, -1, 1}] ]]; a[n_] := Sum[b[n, k], {k, 1, n}]; Array[a, 50] (* Jean-François Alcover, Mar 13 2015, after Alois P. Heinz *)

N=70;  nil=-1;
T = matrix(N, N, i, j, nil);
doIt(last, left) = my(c); c = T[last, left]; if (c == nil, c = 0; for (i = max(1, last - 1), last + 1, c += b(i, left - i)); T[last, left] = c); c;
b(last, left) = if (left == 0, return(1)); if (left < 0, return(0)); doIt(last, left);
a(n) = sum (i = 1, n, b(i, n - i));
vector(N,n,a(n))  \\ David Wasserman, Feb 02 2006

from sympy.core.cache import cacheit
@cacheit
def b(n, i): return 1 if n==i else 0 if n<0 or i<1 else sum(b(n - i, i + j) for j in range(-1, 2))
def a(n): return sum(b(n, k) for k in range(n + 1))
print([a(n) for n in range(51)]) # Indranil Ghosh, Aug 14 2017, after Maple code

a(n) ~ c * d^n, where d = 1.668202067018461116361070469945501401879811945303435230637248..., c = 0.762436680050402638439806786781869262562176911054246754543346... . - Vaclav Kotesovec, Sep 02 2014

More terms from David Wasserman, Feb 02 2006

a(0)=1 prepended by Alois P. Heinz, Aug 14 2017

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

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

A350839 Number of integer partitions of n with a difference < -1 and a conjugate difference < -1.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 3, 7, 11, 17, 26, 39, 54, 81, 108, 148, 201, 269, 353, 467, 601, 779, 995, 1272, 1605, 2029, 2538, 3171, 3941, 4881, 6012, 7405, 9058, 11077, 13478, 16373, 19817, 23953, 28850, 34692, 41599, 49802, 59461, 70905, 84321, 100155, 118694

Offset: 0

Gus Wiseman, Jan 24 2022

nonn

We define a difference of a partition to be a difference of two adjacent parts.

			The a(5) = 1 through a(10) = 17 partitions:
  (311)  (411)   (511)    (422)     (522)      (622)
         (3111)  (4111)   (611)     (711)      (811)
                 (31111)  (3311)    (4221)     (4222)
                          (4211)    (4311)     (4411)
                          (5111)    (5211)     (5221)
                          (41111)   (6111)     (5311)
                          (311111)  (33111)    (6211)
                                    (42111)    (7111)
                                    (51111)    (42211)
                                    (411111)   (43111)
                                    (3111111)  (52111)
                                               (61111)
                                               (331111)
                                               (421111)
                                               (511111)
                                               (4111111)
                                               (31111111)

Allowing -1 gives A144300 = non-constant partitions.
Taking one of the two conditions gives A239955, ranked by A073492, A065201.
These partitions are ranked by A350841.
A000041 = integer partitions, strict A000009.
A034296 = flat (contiguous) partitions, strict A001227.
A073491 = numbers whose prime indices have no gaps, strict A137793.
A090858 = partitions with a single hole, ranked by A325284.
A116931 = partitions with differences != -1, strict A003114.
A116932 = partitions with differences != -1 or -2, strict A025157.
A277103 = partitions with the same number of odd parts as their conjugate.
A350837 = partitions with no adjacent doublings, strict A350840.
A350842 = partitions with differences != -2, strict A350844, sets A005314.
Cf. A000070, A000929, A008284, A040039, A183558, A319630, A321440, A350838.

conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],(Min@@Differences[#]<-1)&&(Min@@Differences[conj[#]]<-1)&]],{n,0,30}]

A350844 Number of strict integer partitions of n with no difference -2.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 3, 4, 4, 7, 7, 8, 11, 12, 15, 18, 21, 23, 31, 32, 40, 45, 54, 59, 73, 78, 94, 106, 122, 136, 161, 177, 203, 231, 259, 293, 334, 372, 417, 476, 525, 592, 663, 742, 821, 931, 1020, 1147, 1271, 1416, 1558, 1752, 1916, 2137, 2357, 2613, 2867

Offset: 0

Gus Wiseman, Jan 21 2022

nonn

			The a(1) = 1 through a(12) = 11 partitions (A..C = 10..12):
  1   2   3    4   5    6     7    8     9     A      B     C
          21       32   51    43   62    54    73     65    84
                   41   321   52   71    63    82     74    93
                              61   521   72    91     83    A2
                                         81    541    92    B1
                                         432   721    A1    543
                                         621   4321   632   651
                                                      821   732
                                                            741
                                                            921
                                                            6321

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

The version for no difference 0 is A000009.
The version for no difference > -2 is A001227, non-strict A034296.
The version for no difference -1 is A003114 (A325160).
The version for subsets of prescribed maximum is A005314.
The version for all differences < -2 is A025157, non-strict A116932.
The opposite version is A072670.
The multiplicative version is A350840, non-strict A350837 (A350838).
The non-strict version is A350842.
A000041 counts integer partitions.
A027187 counts partitions of even length.
A027193 counts partitions of odd length (A026424).
A116931 counts partitions with no difference -1 (A319630).
A323092 counts double-free integer partitions (A320340) strict A120641.
A325534 counts separable partitions (A335433).
A325535 counts inseparable partitions (A335448).
Cf. A000929, A003000, A018819, A040039, A045690, A045691, A154402, A303362, A323094, A342095, A342097.

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

A375136 Number of maximal strictly increasing runs in the weakly increasing prime factors of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Aug 04 2024

nonn

For n > 1, this is one more than the number of adjacent equal terms in the multiset of prime factors of n.

			The prime factors of 540 are {2,2,3,3,3,5}, with maximal strictly increasing runs ({2},{2,3},{3},{3,5}), so a(540) = 4.

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

For compositions we have A124768, row-lengths of A374683, sum A374684.
For sum of prime indices we have A374706.
Row-lengths of A375128.
A112798 lists prime indices:
- distinct A001221
- length A001222
- leader A055396
- sum A056239
- reverse A296150
Cf. A034296, A046660, A066328, A141199, A141809, A279790, A320324, A333213, A358836, A374700.

Table[Length[Split[Flatten[ConstantArray@@@FactorInteger[n]],Less]],{n,100}]

For n > 1, a(n) = A046660(n) + 1 = A001222(n) - A001221(n) + 1.

A342332 Number of compositions of n with all adjacent parts (x, y) satisfying x > 2y or y > 2x.

Original entry on oeis.org

1, 1, 1, 1, 3, 4, 4, 7, 12, 17, 23, 34, 51, 75, 111, 164, 239, 350, 520, 767, 1123, 1652, 2439, 3587, 5263, 7745, 11411, 16789, 24695, 36347, 53489, 78686, 115779, 170390, 250711, 368866, 542783, 798713, 1175208, 1729189, 2544462, 3744077, 5509068, 8106165, 11927785, 17550956, 25824938, 37999743, 55914293, 82274088, 121060721

Offset: 0

Gus Wiseman, Mar 10 2021

nonn

			The a(1) =  1 through a(9) = 17 compositions:
  (1)  (2)  (3)  (4)   (5)    (6)    (7)    (8)     (9)
                 (13)  (14)   (15)   (16)   (17)    (18)
                 (31)  (41)   (51)   (25)   (26)    (27)
                       (131)  (141)  (52)   (62)    (72)
                                     (61)   (71)    (81)
                                     (151)  (152)   (162)
                                     (313)  (161)   (171)
                                            (251)   (252)
                                            (314)   (261)
                                            (413)   (315)
                                            (1313)  (414)
                                            (3131)  (513)
                                                    (1314)
                                                    (1413)
                                                    (3141)
                                                    (4131)
                                                    (13131)

Alois P. Heinz, Table of n, a(n) for n = 0..2500

The unordered version (partitions) is A342098.
Reversing operators and changing 'or' into 'and' gives A342330 (strict: A342341).
The version allowing equality (i.e., non-strict relations) is A342333.
The version allowing partial equality is counted by A342334.
A000929 counts partitions with adjacent parts x >= 2y.
A002843 counts compositions with adjacent parts x <= 2y.
A154402 counts partitions 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).
A342331 counts compositions with adjacent parts x = 2y or y = 2x.
A342335 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.
Cf. A003114, A003242, A034296, A167606, A342083, A342084, A342087, A342191, A342336, A342340.

b:= proc(n, i) option remember; `if`(n=0, 1, add(b(n-j, j),
      j=select(x-> i=0 or x>2*i or i>2*x , {$1..n})))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, May 24 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, Select[Range[n], i == 0 || # > 2 i || i > 2 # &]}]];
a[n_] := b[n, 0];
a /@ Range[0, 50] (* Jean-François Alcover, Jun 09 2021, after Alois P. Heinz *)

More terms from Joerg Arndt, Mar 12 2021

cp's OEIS Frontend

A239955 Number of partitions p of n such that (number of distinct parts of p) <= max(p) - min(p).

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A375128 Irregular triangle read by rows where row n lists the minima of maximal strictly increasing runs in the weakly increasing prime indices of n.

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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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A034297 Number of ordered positive integer solutions (m_1, m_2, ..., m_k) (for some k) to Sum_{i=1..k} m_i=n with |m_i-m_{i-1}| <= 1 for i = 2 ... k.

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

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A350839 Number of integer partitions of n with a difference < -1 and a conjugate difference < -1.

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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).