A047967 -id:A047967 - cp's OEIS frontend

A381438 Triangle read by rows where T(n>0,k>0) is the number of integer partitions of n whose section-sum partition ends with k.

1, 1, 1, 1, 0, 2, 2, 1, 0, 2, 3, 1, 0, 0, 3, 4, 1, 2, 0, 0, 4, 7, 2, 1, 0, 0, 0, 5, 9, 4, 1, 2, 0, 0, 0, 6, 13, 4, 4, 1, 0, 0, 0, 0, 8, 18, 6, 3, 2, 3, 0, 0, 0, 0, 10, 26, 9, 5, 2, 2, 0, 0, 0, 0, 0, 12, 32, 12, 8, 4, 2, 4, 0, 0, 0, 0, 0, 15

Offset: 1

Gus Wiseman, Mar 01 2025

The section-sum partition (A381436) of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

			Triangle begins:
   1
   1  1
   1  0  2
   2  1  0  2
   3  1  0  0  3
   4  1  2  0  0  4
   7  2  1  0  0  0  5
   9  4  1  2  0  0  0  6
  13  4  4  1  0  0  0  0  8
  18  6  3  2  3  0  0  0  0 10
  26  9  5  2  2  0  0  0  0  0 12
  32 12  8  4  2  4  0  0  0  0  0 15
  47 16 11  4  3  2  0  0  0  0  0  0 18
  60 23 12  8  3  2  5  0  0  0  0  0  0 22
  79 27 20  7  9  4  3  0  0  0  0  0  0  0 27
 Row n = 9 counts the following partitions:
  (711)        (522)    (333)     (441)  .  .  .  .  (9)
  (6111)       (4221)   (3321)                       (81)
  (5211)       (3222)   (32211)                      (72)
  (51111)      (22221)  (222111)                     (63)
  (4311)                                             (621)
  (42111)                                            (54)
  (411111)                                           (531)
  (33111)                                            (432)
  (321111)
  (3111111)
  (2211111)
  (21111111)
  (111111111)

Last column (k=n) is A000009.
Row sums are A000041.
Row sums without the last column (k=n) are A047967.
For first instead of last part we have A116861, rank A066328.
First column (k=1) is A241131 shifted right and starting with 1 instead of 0.
Using Heinz numbers, this statistic is given by A381437.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts section-sum partitions, complement A351293.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Section-sum partition: A381431, A381432, A381433, A381434, A381435, A381436.
Look-and-Say partition: A048767, A351294, A351295, A381440.
Cf. A047966, A051903, A051904, A091602, A181819, A212166.

egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
Table[Length[Select[IntegerPartitions[n],k==Last[egs[#]]&]],{n,15},{k,n}]

A006477 Number of partitions of n with at least 1 odd and 1 even part.

Original entry on oeis.org

0, 0, 0, 1, 1, 4, 4, 10, 11, 22, 25, 44, 51, 83, 98, 149, 177, 259, 309, 436, 521, 716, 857, 1151, 1376, 1816, 2170, 2818, 3361, 4309, 5132, 6502, 7728, 9695, 11501, 14298, 16924, 20877, 24661, 30203, 35598, 43323, 50956, 61651, 72357, 87086, 101999

Offset: 0

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
M. O. LeVan, A triangle for partitions, Amer. Math. Monthly, 79 (1972), 507-510.

Cf. A047967, A038348.

a[n_?OddQ] := PartitionsP[n] - PartitionsQ[n]; a[n_?EvenQ] := PartitionsP[n] - PartitionsQ[n] - PartitionsP[n/2]; a[0] = 0; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Mar 17 2014, after Vladeta Jovovic *)

Convolution of 0, 1, 1, 2, 2, 3, 4, 5, 6, ... (essentially A000009) and 0, 0, 1, 0, 2, 0, 3, 0, 5, ... (essentially A035363).
G.f.: (prod(1/(1-x^k), k odd)-1) * (prod(1/(1-x^k), k even)-1).
A000041(n)-A000009(n) if n is odd else A000041(n)-A000009(n)-A000041(n/2). - Vladeta Jovovic, Sep 10 2003
a(n) = A000041(n) - A096441(n), n >= 1. - Omar E. Pol, Aug 16 2013

More terms from David W. Wilson, May 11 2001

A238354 Triangle T(n,k) read by rows: T(n,k) is the number of partitions of n (as weakly ascending list of parts) with minimal ascent k, n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 2, 1, 0, 0, 4, 0, 1, 0, 0, 5, 1, 0, 1, 0, 0, 8, 1, 1, 0, 1, 0, 0, 11, 2, 0, 1, 0, 1, 0, 0, 17, 2, 1, 0, 1, 0, 1, 0, 0, 23, 3, 1, 1, 0, 1, 0, 1, 0, 0, 33, 4, 2, 0, 1, 0, 1, 0, 1, 0, 0, 45, 5, 2, 1, 0, 1, 0, 1, 0, 1, 0, 0, 63, 6, 3, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 84, 8, 3, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 114, 10, 4, 2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0

Offset: 0

Joerg Arndt and Alois P. Heinz, Feb 26 2014

Column k=0: T(n,0) = 1 + A047967(n).
Column k=1 is A238708.
Row sums are A000041.

			Triangle starts:
  00:    1;
  01:    1,  0;
  02:    2,  0, 0;
  03:    2,  1, 0, 0;
  04:    4,  0, 1, 0, 0;
  05:    5,  1, 0, 1, 0, 0;
  06:    8,  1, 1, 0, 1, 0, 0;
  07:   11,  2, 0, 1, 0, 1, 0, 0;
  08:   17,  2, 1, 0, 1, 0, 1, 0, 0;
  09:   23,  3, 1, 1, 0, 1, 0, 1, 0, 0;
  10:   33,  4, 2, 0, 1, 0, 1, 0, 1, 0, 0;
  11:   45,  5, 2, 1, 0, 1, 0, 1, 0, 1, 0, 0;
  12:   63,  6, 3, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0;
  13:   84,  8, 3, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0;
  14:  114, 10, 4, 2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0;
  15:  150, 13, 4, 3, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0;
  ...
The 11 partitions of 6 together with their minimal ascents are:
  01:  [ 1 1 1 1 1 1 ]   0
  02:  [ 1 1 1 1 2 ]     0
  03:  [ 1 1 1 3 ]       0
  04:  [ 1 1 2 2 ]       0
  05:  [ 1 1 4 ]         0
  06:  [ 1 2 3 ]         1
  07:  [ 1 5 ]           4
  08:  [ 2 2 2 ]         0
  09:  [ 2 4 ]           2
  10:  [ 3 3 ]           0
  11:  [ 6 ]             0
There are 8 partitions of 6 with min ascent 0, 1 with min ascents 1, 2, and 4, giving row 6 of the triangle: 8, 1, 1, 0, 1, 0, 0.

Joerg Arndt and Alois P. Heinz, Rows n = 0..140, flattened

Cf. A238353 (partitions by maximal ascent).

b:= proc(n, i, t) option remember; `if`(n=0, 1/x, `if`(i<1, 0,
      b(n, i-1, t)+`if`(i>n, 0, (p->`if`(t=0, p, add(coeff(
       p, x, j)*x^`if`(j<0, t-i, min(j, t-i)),
       j=-1..degree(p))))(b(n-i, i, i)))))
    end:
T:= n->(p->seq(coeff(p, x, k)+`if`(k=0, 1, 0), k=0..n))(b(n$2, 0)):
seq(T(n), n=0..15);

b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1/x, If[i<1, 0, b[n, i-1, t]+If[i>n, 0, Function[{p}, If[t == 0, p, Sum[Coefficient[p, x, j]*x^If[j<0, t-i, Min[j, t-i]], {j, -1, Exponent[p, x]}]]][b[n-i, i, i]]]]]; T[n_] := Function[{p}, Table[ Coefficient[p, x, k]+If[k == 0, 1, 0], {k, 0, n}]][b[n, n, 0]]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Jan 12 2015, translated from Maple *)

A366845 Number of integer partitions of n that contain at least one even part and whose halved even parts are relatively prime.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 23, 31, 43, 58, 82, 107, 144, 189, 250, 323, 420, 537, 695, 880, 1114, 1404, 1774, 2210, 2759, 3423, 4239, 5223, 6430, 7869, 9640, 11738, 14266, 17297, 20950, 25256, 30423, 36545, 43824, 52421, 62620, 74599, 88802, 105431

Offset: 0

Gus Wiseman, Oct 28 2023

nonn

			The partition y = (6,4) has halved even parts (3,2) which are relatively prime, so y is counted under a(10).
The a(2) = 1 through a(9) = 15 partitions:
  (2)  (21)  (22)   (32)    (42)     (52)      (62)       (72)
             (211)  (221)   (222)    (322)     (332)      (432)
                    (2111)  (321)    (421)     (422)      (522)
                            (2211)   (2221)    (521)      (621)
                            (21111)  (3211)    (2222)     (3222)
                                     (22111)   (3221)     (3321)
                                     (211111)  (4211)     (4221)
                                               (22211)    (5211)
                                               (32111)    (22221)
                                               (221111)   (32211)
                                               (2111111)  (42111)
                                                          (222111)
                                                          (321111)
                                                          (2211111)
                                                          (21111111)

For all parts we have A000837, complement A018783.
These partitions have ranks A366847.
For odd parts we have A366850, ranks A366846, complement A366842.
A000041 counts integer partitions, strict A000009, complement A047967.
A035363 counts partitions into all even parts, ranks A066207.
A078374 counts relatively prime strict partitions.
A168532 counts partitions by gcd.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
A366531 = 2*A366533 adds up even prime indices, triangle A113686/A174713.
Cf. A051424, A113685, A116598, A258117, A289509, A366843, A366849.

Table[Length[Select[IntegerPartitions[n], GCD@@Select[#,EvenQ]/2==1&]],{n,0,30}]

A194543 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) is the number of partitions of n into parts p_i such that |p_i - p_j| >= k for i != j.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 5, 2, 2, 1, 1, 7, 3, 2, 2, 1, 1, 11, 4, 3, 2, 2, 1, 1, 15, 5, 3, 3, 2, 2, 1, 1, 22, 6, 4, 3, 3, 2, 2, 1, 1, 30, 8, 5, 4, 3, 3, 2, 2, 1, 1, 42, 10, 6, 4, 4, 3, 3, 2, 2, 1, 1, 56, 12, 7, 5, 4, 4, 3, 3, 2, 2, 1, 1, 77, 15, 9, 6, 5, 4, 4, 3, 3, 2, 2, 1, 1

Offset: 0

Alois P. Heinz, Aug 29 2011

T(n,k) = 1 for n >= 0 and k >= n.

In general, column k > 0 is asymptotic to c^(1/4) * r * exp(2*sqrt(c*n)) / (2*sqrt(Pi*(1-r)*(r + k*(1-r))) * n^(3/4)), where r is the smallest real root of the equation r^k + r = 1 and c = k*log(r)^2/2 + polylog(2, 1-r). - Vaclav Kotesovec, Jan 02 2016

			T(7,3) = 3: [7], [6,1], [5,2].
T(23,6) = 11: [23], [22,1], [21,2], [20,3], [19,4], [18,5], [17,6], [16,7], [15,8], [15,7,1], [14,8,1].
Triangle begins:
   1;
   1, 1;
   2, 1, 1;
   3, 2, 1, 1;
   5, 2, 2, 1, 1;
   7, 3, 2, 2, 1, 1;
  11, 4, 3, 2, 2, 1, 1;
  15, 5, 3, 3, 2, 2, 1, 1;

Alois P. Heinz, Rows n = 0..140

Columns 0-8 give: A000041, A000009, A003114, A025157, A025158, A025159, A025160, A025161, A025162. T(n,0)-T(n,1) = A047967(n).

b:= proc(n, i, k) option remember;
      if n<0 then 0
    elif n=0 then 1
    else add(b(n-i-j, i+j, k), j=k..n-i)
      fi
    end:
T:= (n, k)-> `if`(n=0, 1, 0) +add(b(n-i, i, k), i=1..n):
seq(seq(T(n, k), k=0..n), n=0..20);

b[n_, i_, k_] := b[n, i, k] = If[n<0, 0, If[n == 0, 1, Sum[b[n-i-j, i+j, k], {j, k, n-i}]]]; T[n_, k_] := If[n == 0, 1, 0] + Sum[b[n-i, i, k], {i, 1, n}]; Table[ Table[T[n, k], {k, 0, n}], {n, 0, 20}] // Flatten (* Jean-François Alcover, Jan 19 2015, after Alois P. Heinz *)

G.f. of column k: Sum_{j>=0} x^(j*((j-1)*k/2+1))/Product_{i=1..j} (1-x^i).

A366850 Number of integer partitions of n whose odd parts are relatively prime.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 7, 11, 16, 22, 32, 43, 60, 80, 110, 140, 194, 244, 327, 410, 544, 670, 883, 1081, 1401, 1708, 2195, 2651, 3382, 4069, 5129, 6157, 7708, 9194, 11438, 13599, 16788, 19911, 24432, 28858, 35229, 41507, 50359, 59201, 71489, 83776, 100731, 117784

Offset: 0

Gus Wiseman, Oct 28 2023

nonn

			The a(1) = 1 through a(8) = 16 partitions:
  (1)  (11)  (21)   (31)    (41)     (51)      (61)       (53)
             (111)  (211)   (221)    (321)     (331)      (71)
                    (1111)  (311)    (411)     (421)      (431)
                            (2111)   (2211)    (511)      (521)
                            (11111)  (3111)    (2221)     (611)
                                     (21111)   (3211)     (3221)
                                     (111111)  (4111)     (3311)
                                               (22111)    (4211)
                                               (31111)    (5111)
                                               (211111)   (22211)
                                               (1111111)  (32111)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

For all parts (not just odd) we have A000837, complement A018783.
The complement is counted by A366842.
These partitions have ranks A366846.
A000041 counts integer partitions, strict A000009 (also into odds).
A000740 counts relatively prime compositions.
A078374 counts relatively prime strict partitions.
A113685 counts partitions by sum of odd parts, rank statistic A366528.
A168532 counts partitions by gcd.
A239261 counts partitions with (sum of odd parts) = (sum of even parts).
Cf. A007359, A047967, A051424, A066208, A116598, A365067, A366843, A366844, A366845, A366848.

Table[Length[Select[IntegerPartitions[n],GCD@@Select[#,OddQ]==1&]],{n,0,30}]

A167934 a(n) = A000041(n) - A032741(n).

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 8, 14, 19, 28, 39, 55, 72, 100, 132, 173, 227, 296, 380, 489, 622, 789, 999, 1254, 1568, 1956, 2433, 3007, 3713, 4564, 5597, 6841, 8344, 10140, 12307, 14880, 17969, 21636, 26012, 31182, 37331, 44582, 53167, 63260, 75170

Offset: 0

Omar E. Pol, Nov 16 2009

nonn

a(n) is also the number of partitions of n whose parts are not all equal, (including however the partition with a single part of size n). Note that the number of partitions of n whose parts are all equal gives the number of divisors of n, for n>0. (See also A144300.)

			The partitions of n = 6 are:
6 ....................... All parts are equal, but included .. (1).
5 + 1 ................... All parts are not equal ............ (2).
4 + 2 ................... All parts are not equal ............ (3).
4 + 1 + 1 ............... All parts are not equal ............ (4).
3 + 3 ................... All parts are equal, not included.
3 + 2 + 1 ............... All parts are not equal ............ (5).
3 + 1 + 1 + 1 ........... All parts are not equal ............ (6).
2 + 2 + 2 ............... All parts are equal, not included.
2 + 2 + 1 + 1 ........... All parts are not equal ............ (7).
2 + 1 + 1 + 1 + 1 ....... All parts are not equal ............ (8).
1 + 1 + 1 + 1 + 1 + 1 ... All parts are equal, not included.
Then a(6) = 8.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Omar E. Pol, Illustration of the shell model of partitions (2D and 3D view)
Omar E. Pol, Illustration of the shell model of partitions (2D view)
Omar E. Pol, Illustration of the shell model of partitions (3D view)

Cf. A000005, A000009, A000041, A000065, A032741, A047967, A111133, A144300, A135010, A138121, A167930, A167932, A167935.

b:= proc(n, i, k) option remember;
      if n<0 then 0
    elif n=0 then `if`(k=0, 1, 0)
    elif i=0 then 0
    else b(n, i-1, k)+
         b(n-i, i, `if`(k<0, i, `if`(k<>i, 0, k)))
      fi
    end:
a:= n-> 1 +b(n, n-1, -1):
seq(a(n), n=0..50);  #  Alois P. Heinz, Dec 01 2010

a[0] = 1; a[n_] := PartitionsP[n] - DivisorSigma[0, n] + 1; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jan 08 2016 *)

a(n) = A000041(n) - A032741(n).

More terms from Alois P. Heinz, Dec 01 2010

A194452 Total number of repeated parts in all partitions of n.

Original entry on oeis.org

0, 0, 2, 3, 8, 12, 24, 35, 60, 87, 136, 192, 287, 396, 567, 773, 1074, 1439, 1958, 2587, 3454, 4514, 5931, 7666, 9951, 12736, 16341, 20743, 26354, 33184, 41807, 52262, 65329, 81144, 100721, 124344, 153390, 188303, 230940, 282063, 344100, 418242, 507762

Offset: 0

Omar E. Pol, Nov 19 2011

nonn

			For n = 6 we have:
--------------------------------------
.                        Number of
Partitions             repeated parts
--------------------------------------
6 .......................... 0
3 + 3 ...................... 2
4 + 2 ...................... 0
2 + 2 + 2 .................. 3
5 + 1 ...................... 0
3 + 2 + 1 .................. 0
4 + 1 + 1 .................. 2
2 + 2 + 1 + 1 .............. 4
3 + 1 + 1 + 1 .............. 3
2 + 1 + 1 + 1 + 1 .......... 4
1 + 1 + 1 + 1 + 1 + 1 ...... 6
------------------------------------
Total ..................... 24
So a(6) = 24.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A006128, A024786, A047967, A135010, A138121, A194544, A264405.

b:= proc(n, i) option remember; local h, j, t;
      if n<0 then [0, 0]
    elif n=0 then [1, 0]
    elif i<1 then [0, 0]
    else h:= [0, 0];
         for j from 0 to iquo(n, i) do
           t:= b(n-i*j, i-1);
           h:= [h[1]+t[1], h[2]+t[2]+`if`(j<2, 0, t[1]*j)]
         od; h
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..50);  # Alois P. Heinz, Nov 20 2011
g := add(x^(2*j)*(2-x^j)/(1-x^j), j = 1 .. 80)/mul(1-x^j, j = 1 .. 80): gser := series(g, x = 0, 50): seq(coeff(gser, x, n), n = 0 .. 45); # Emeric Deutsch, Feb 02 2016

myCount[p_List] := Module[{t}, If[p == {}, 0, t = Transpose[Tally[p]][[2]]; Sum[If[t[[i]] == 1, 0, t[[i]]], {i, Length[t]}]]]; Table[Total[Table[myCount[p], {p, IntegerPartitions[i]}]], {i, 0, 20}] (* T. D. Noe, Nov 19 2011 *)
b[n_, i_] := b[n, i] = Module[{h, j, t}, Which[n<0, {0, 0}, n==0, {1, 0}, i < 1, {0, 0}, True, h={0, 0}; For[j=0, j <= Quotient[n, i], j++, t = b[n - i*j, i-1]; h = {h[[1]]+t[[1]], h[[2]]+t[[2]] + If[j<2, 0, t[[1]]*j]}]; h] ]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Oct 25 2015, after Alois P. Heinz *)
Table[Length[Flatten[Select[Flatten[Split[#]&/@IntegerPartitions[n],1],Length[#]>1&]]],{n,0,60}] (* Harvey P. Dale, Jun 12 2024 *)

a(n) = A006128(n) - A024786(n+1).
a(n) = Sum_{k=2..n} k*A264405(n,k). - Alois P. Heinz, Dec 07 2015
G.f.: g = Sum_{j>0} (x^{2*j}*(2 - x^j)/(1-x^j))/Product_{k>0}(1 - x^k) (obtained by logarithmic differentiation of the bivariate g.f. given in A264405). - Emeric Deutsch, Feb 02 2016

A321773 Number of compositions of n into parts with distinct multiplicities and with exactly three parts.

Original entry on oeis.org

1, 3, 6, 4, 9, 9, 10, 12, 15, 13, 18, 18, 19, 21, 24, 22, 27, 27, 28, 30, 33, 31, 36, 36, 37, 39, 42, 40, 45, 45, 46, 48, 51, 49, 54, 54, 55, 57, 60, 58, 63, 63, 64, 66, 69, 67, 72, 72, 73, 75, 78, 76, 81, 81, 82, 84, 87, 85, 90, 90, 91, 93, 96, 94, 99, 99

Offset: 3

Alois P. Heinz, Nov 18 2018

nonn

			From _Gus Wiseman_, Nov 11 2020: (Start)
Also the number of 3-part non-strict compositions of n. For example, the a(3) = 1 through a(11) = 15 triples are:
  111   112   113   114   115   116   117   118   119
        121   122   141   133   161   144   181   155
        211   131   222   151   224   171   226   191
              212   411   223   233   225   244   227
              221         232   242   252   262   272
              311         313   323   333   334   335
                          322   332   414   343   344
                          331   422   441   424   353
                          511   611   522   433   434
                                      711   442   443
                                            622   515
                                            811   533
                                                  551
                                                  722
                                                  911
(End)

Alois P. Heinz, Table of n, a(n) for n = 3..1000

Column k=3 of A242887.
A235451 counts 3-part compositions with distinct run-lengths
A001399(n-6) counts 3-part compositions in the complement.
A014311 intersected with A335488 ranks these compositions.
A140106 is the unordered case, with Heinz numbers A285508.
A261982 counts non-strict compositions of any length.
A001523 counts unimodal compositions, with complement A115981.
A007318 and A097805 count compositions by length.
A032020 counts strict compositions.
A047967 counts non-strict partitions, with Heinz numbers A013929.
A242771 counts triples that are not strictly increasing.
Cf. A000212, A000217, A001840, A128422, A156040, A332834, A337461, A337484, A337603, A337604.

Table[Length[Join@@Permutations/@Select[IntegerPartitions[n,{3}],!UnsameQ@@#&]],{n,0,100}] (* Gus Wiseman, Nov 11 2020 *)

Conjectures from Colin Barker, Dec 11 2018: (Start)
G.f.: x^3*(1 + 3*x + 5*x^2) / ((1 - x)^2*(1 + x)*(1 + x + x^2)).
a(n) = a(n-2) + a(n-3) - a(n-5) for n>7. (End)
Conjecture: a(n) = (3*n-k)/2 where k value has a cycle of 6 starting from n=3 of (7,6,3,10,3,6). - Bill McEachen, Aug 12 2025

A350140 Nonsquarefree numbers whose prime signature has at least one odd part other the first or last.

Original entry on oeis.org

60, 84, 120, 132, 140, 150, 156, 168, 204, 220, 228, 240, 260, 264, 270, 276, 280, 294, 300, 308, 312, 315, 336, 340, 348, 364, 372, 378, 380, 408, 420, 440, 444, 456, 460, 476, 480, 490, 492, 495, 516, 520, 528, 532, 540, 552, 560, 564, 572, 580, 585, 588

Offset: 1

Gus Wiseman, Dec 25 2021

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

Also Heinz numbers of non-weakly alternating non-strict integer partitions, where we define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. These partitions are counted by A349796. This sequence involves the somewhat degenerate case where no strict increases are allowed.

			The terms together with their Heinz partitions begin (A-E = 10-14):
     60: (3211)      276: (9211)      420: (43211)
     84: (4211)      280: (43111)     440: (53111)
    120: (32111)     294: (4421)      444: (C211)
    132: (5211)      300: (33211)     456: (82111)
    140: (4311)      308: (5411)      460: (9311)
    150: (3321)      312: (62111)     476: (7411)
    156: (6211)      315: (4322)      480: (3211111)
    168: (42111)     336: (421111)    490: (4431)
    204: (7211)      340: (7311)      492: (D211)
    220: (5311)      348: (A211)      495: (5322)
    228: (8211)      364: (6411)      516: (E211)
    240: (321111)    372: (B211)      520: (63111)
    260: (6311)      378: (42221)     528: (521111)
    264: (52111)     380: (8311)      532: (8411)
    270: (32221)     408: (72111)     540: (322211)

Including all nonsquarefree numbers gives A013929, complement A005117.
Subsets include A088860 and A110286.
Signatures of this type are counted by A274230, complement A027383.
The strict instead of non-strict version is A336568, counted by A347548.
A version for compositions allowing strict is A349057, counted by A349053.
Allowing strict partitions gives A349794, counted by A349061.
These partitions are counted by A349796.
The complement in nonsquarefree partitions is A350137, counted by A349795.
A000041 = integer partitions, strict A000009.
A001250 = alternating permutations, ranked by A349051, complement A348615.
A003242 = Carlitz (anti-run) compositions.
A025047/A025048/A025049 = alternating compositions, ranked by A345167.
A056239 adds up prime indices, row sums of A112798, row lengths A001222.
A096441 = weakly alternating 0-appended partitions.
A124010 = prime signature, sorted A118914.
A345164 = alternating permutations of prime indices, complement A350251.
A345170 = partitions w/ an alternating permutation, ranked by A345172.
A349052/A129852/A129853 = weakly alternating compositions.
A349056 = weakly alternating permutations of prime indices.
A349058 = weakly alternating patterns, complement A350138.
A349060 = weakly alternating partitions, strong A349801.
A349798 = weakly but not strongly alternating perms of prime indices.
Cf. A000111, A047967, A333213, A335448, A344615, A344653, A345173, A349054, A349059, A349797, A349799.

Select[Range[300],!SquareFreeQ[#]&&PrimeNu[#]>1&& !And@@EvenQ/@Take[Last/@FactorInteger[#],{2,-2}]&]

Complement of A005117 in A349794.

cp's OEIS Frontend

A381438 Triangle read by rows where T(n>0,k>0) is the number of integer partitions of n whose section-sum partition ends with k.

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A006477 Number of partitions of n with at least 1 odd and 1 even part.

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A238354 Triangle T(n,k) read by rows: T(n,k) is the number of partitions of n (as weakly ascending list of parts) with minimal ascent k, n >= 0, 0 <= k <= n.

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A366845 Number of integer partitions of n that contain at least one even part and whose halved even parts are relatively prime.

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A194543 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) is the number of partitions of n into parts p_i such that |p_i - p_j| >= k for i != j.

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A366850 Number of integer partitions of n whose odd parts are relatively prime.

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A167934 a(n) = A000041(n) - A032741(n).

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A194452 Total number of repeated parts in all partitions of n.

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