A035377 -id:A035377 - cp's OEIS frontend

A035363 Number of partitions of n into even parts.

1, 0, 1, 0, 2, 0, 3, 0, 5, 0, 7, 0, 11, 0, 15, 0, 22, 0, 30, 0, 42, 0, 56, 0, 77, 0, 101, 0, 135, 0, 176, 0, 231, 0, 297, 0, 385, 0, 490, 0, 627, 0, 792, 0, 1002, 0, 1255, 0, 1575, 0, 1958, 0, 2436, 0, 3010, 0, 3718, 0, 4565, 0, 5604, 0, 6842, 0, 8349, 0, 10143, 0, 12310, 0

Offset: 0

Olivier Gérard

Convolved with A036469 = A000070. - Gary W. Adamson, Jun 09 2009
Note that these partitions are located in the head of the last section of the set of partitions of n (see A135010). - Omar E. Pol, Nov 20 2009
Number of symmetric unimodal compositions of n+2 where the maximal part appears twice, see example. Also number of symmetric unimodal compositions of n where the maximal part appears an even number of times. - Joerg Arndt, Jun 11 2013
Number of partitions of n having parts of even multiplicity. These are the conjugates of the partitions from the definition. Example: a(8)=5 because we have [4,4],[3,3,1,1],[2,2,2,2],[2,2,1,1,1,1], and [1,1,1,1,1,1,1,1]. - Emeric Deutsch, Jan 27 2016
From Gus Wiseman, May 22 2021: (Start)
The Heinz numbers of the conjugate partitions described in Emeric Deutsch's comment above are given by A000290.
For n > 1, also the number of integer partitions of n-1 whose only odd part is the smallest. The Heinz numbers of these partitions are given by A341446. For example, the a(2) = 1 through a(14) = 15 partitions (empty columns shown as dots, A..D = 10..13) are:
  1  .  3   .  5    .  7     .  9      .  B       .  D
        21     41      43       63        65         85
               221     61       81        83         A3
                       421      441       A1         C1
                       2221     621       443        643
                                4221      641        661
                                22221     821        841
                                          4421       A21
                                          6221       4441
                                          42221      6421
                                          222221     8221
                                                     44221
                                                     62221
                                                     422221
                                                     2222221
Also the number of integer partitions of n whose greatest part is the sum of all the other parts. The Heinz numbers of these partitions are given by A344415. For example, the a(2) = 1 through a(12) = 11 partitions (empty columns not shown) are:
  (11)  (22)   (33)    (44)     (55)      (66)
        (211)  (321)   (422)    (532)     (633)
               (3111)  (431)    (541)     (642)
                       (4211)   (5221)    (651)
                       (41111)  (5311)    (6222)
                                (52111)   (6321)
                                (511111)  (6411)
                                          (62211)
                                          (63111)
                                          (621111)
                                          (6111111)
Also the number of integer partitions of n of length n/2. The Heinz numbers of these partitions are given by A340387. For example, the a(2) = 1 through a(14) = 15 partitions (empty columns not shown) are:
  (2)  (22)  (222)  (2222)  (22222)  (222222)  (2222222)
       (31)  (321)  (3221)  (32221)  (322221)  (3222221)
             (411)  (3311)  (33211)  (332211)  (3322211)
                    (4211)  (42211)  (333111)  (3332111)
                    (5111)  (43111)  (422211)  (4222211)
                            (52111)  (432111)  (4322111)
                            (61111)  (441111)  (4331111)
                                     (522111)  (4421111)
                                     (531111)  (5222111)
                                     (621111)  (5321111)
                                     (711111)  (5411111)
                                               (6221111)
                                               (6311111)
                                               (7211111)
                                               (8111111)
(End)

			From _Joerg Arndt_, Jun 11 2013: (Start)
There are a(12)=11 symmetric unimodal compositions of 12+2=14 where the maximal part appears twice:
01:  [ 1 1 1 1 1 2 2 1 1 1 1 1 ]
02:  [ 1 1 1 1 3 3 1 1 1 1 ]
03:  [ 1 1 1 4 4 1 1 1 ]
04:  [ 1 1 2 3 3 2 1 1 ]
05:  [ 1 1 5 5 1 1 ]
06:  [ 1 2 4 4 2 1 ]
07:  [ 1 6 6 1 ]
08:  [ 2 2 3 3 2 2 ]
09:  [ 2 5 5 2 ]
10:  [ 3 4 4 3 ]
11:  [ 7 7 ]
There are a(14)=15 symmetric unimodal compositions of 14 where the maximal part appears an even number of times:
01:  [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 2 2 1 1 1 1 1 ]
03:  [ 1 1 1 1 3 3 1 1 1 1 ]
04:  [ 1 1 1 2 2 2 2 1 1 1 ]
05:  [ 1 1 1 4 4 1 1 1 ]
06:  [ 1 1 2 3 3 2 1 1 ]
07:  [ 1 1 5 5 1 1 ]
08:  [ 1 2 2 2 2 2 2 1 ]
09:  [ 1 2 4 4 2 1 ]
10:  [ 1 3 3 3 3 1 ]
11:  [ 1 6 6 1 ]
12:  [ 2 2 3 3 2 2 ]
13:  [ 2 5 5 2 ]
14:  [ 3 4 4 3 ]
15:  [ 7 7 ]
(End)
a(8)=5 because we  have [8], [6,2], [4,4], [4,2,2], and [2,2,2,2]. - _Emeric Deutsch_, Jan 27 2016
From _Gus Wiseman_, May 22 2021: (Start)
The a(0) = 1 through a(12) = 11 partitions into even parts are the following (empty columns shown as dots, A = 10, C = 12). The Heinz numbers of these partitions are given by A066207.
  ()  .  (2)  .  (4)   .  (6)    .  (8)     .  (A)      .  (C)
                 (22)     (42)      (44)       (64)        (66)
                          (222)     (62)       (82)        (84)
                                    (422)      (442)       (A2)
                                    (2222)     (622)       (444)
                                               (4222)      (642)
                                               (22222)     (822)
                                                           (4422)
                                                           (6222)
                                                           (42222)
                                                           (222222)
(End)

Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education, Vol. 31, No. 1, pp. 24-28, Winter 1997. MathEduc Database (Zentralblatt MATH, 1997c.01891).
Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem II, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17. Zentralblatt MATH, Zbl 1071.05501.

Robert Price, Table of n, a(n) for n = 0..2001

Bisection (even part) gives the partition numbers A000041.
Column k=0 of A103919, A264398.
Cf. A036469, A000070.
Cf. A135010, A138121.
Note: A-numbers of ranking sequences are in parentheses below.
The version for odd instead of even parts is A000009 (A066208).
The version for parts divisible by 3 instead of 2 is A035377.
The strict case is A035457.
The Heinz numbers of these partitions are given by A066207.
The ordered version (compositions) is A077957 prepended by (1,0).
This is column k = 2 of A168021.
The multiplicative version (factorizations) is A340785.
A000569 counts graphical partitions (A320922).
A004526 counts partitions of length 2 (A001358).
A025065 counts palindromic partitions (A265640).
A027187 counts partitions with even length/maximum (A028260/A244990).
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A236913 counts partitions of even length and sum (A340784).
A340601 counts partitions of even rank (A340602).
The following count partitions of even length:
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
- A339560 can be partitioned into distinct strict pairs (A339561).
Cf. A000041, A000290, A087897, A100484, A110618, A209816, A210249, A233771, A339004, A340385, A340387, A340786, A341447.

ZL:= [S, {C = Cycle(B), S = Set(C), E = Set(B), B = Prod(Z,Z)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=0..69); # Zerinvary Lajos, Mar 26 2008
g := 1/mul(1-x^(2*k), k = 1 .. 100): gser := series(g, x = 0, 80): seq(coeff(gser, x, n), n = 0 .. 78); # Emeric Deutsch, Jan 27 2016
# Using the function EULER from Transforms (see link at the bottom of the page).
[1,op(EULER([0,1,seq(irem(n,2),n=0..66)]))]; # Peter Luschny, Aug 19 2020
# next Maple program:
a:= n-> `if`(n::odd, 0, combinat[numbpart](n/2)):
seq(a(n), n=0..84);  # Alois P. Heinz, Jun 22 2021

nmax = 50; s = Range[2, nmax, 2];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Aug 05 2020 *)

from sympy import npartitions
def A035363(n): return 0 if n&1 else npartitions(n>>1) # Chai Wah Wu, Sep 23 2023

G.f.: Product_{k even} 1/(1 - x^k).
Convolution with the number of partitions into distinct parts (A000009, which is also number of partitions into odd parts) gives the number of partitions (A000041). - Franklin T. Adams-Watters, Jan 06 2006
If n is even then a(n)=A000041(n/2) otherwise a(n)=0. - Omar E. Pol, Nov 20 2009
G.f.: 1 + x^2*(1 - G(0))/(1-x^2) where G(k) =  1 - 1/(1-x^(2*k+2))/(1-x^2/(x^2-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 23 2013
a(n) = A096441(n) - A000009(n), n >= 1. - Omar E. Pol, Aug 16 2013
G.f.: exp(Sum_{k>=1} x^(2*k)/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Aug 13 2018

A096441 Number of palindromic and unimodal compositions of n. Equivalently, the number of orbits under conjugation of even nilpotent n X n matrices.

Original entry on oeis.org

1, 2, 2, 4, 3, 7, 5, 11, 8, 17, 12, 26, 18, 37, 27, 54, 38, 76, 54, 106, 76, 145, 104, 199, 142, 266, 192, 357, 256, 472, 340, 621, 448, 809, 585, 1053, 760, 1354, 982, 1740, 1260, 2218, 1610, 2818, 2048, 3559, 2590, 4485, 3264, 5616, 4097, 7018, 5120, 8728, 6378

Offset: 1

Nolan R. Wallach (nwallach(AT)ucsd.edu), Aug 10 2004

nonn

Number of partitions of n such that all differences between successive parts are even, see example. [Joerg Arndt, Dec 27 2012]
Number of partitions of n where either all parts are odd or all parts are even. - Omar E. Pol, Aug 16 2013
From Gus Wiseman, Jan 13 2022: (Start)
Also the number of integer partitions of n with all even multiplicities (or run-lengths) except possibly the first. These are the conjugates of the partitions described by Joerg Arndt above. For example, the a(1) = 1 through a(8) = 11 partitions are:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (322)      (44)
                    (211)   (11111)  (222)     (511)      (422)
                    (1111)           (411)     (31111)    (611)
                                     (2211)    (1111111)  (2222)
                                     (21111)              (3311)
                                     (111111)             (22211)
                                                          (41111)
                                                          (221111)
                                                          (2111111)
                                                          (11111111)
(End)

			From _Joerg Arndt_, Dec 27 2012: (Start)
There are a(10)=17 partitions of 10 where all differences between successive parts are even:
[ 1]  [ 1 1 1 1 1 1 1 1 1 1 ]
[ 2]  [ 2 2 2 2 2 ]
[ 3]  [ 3 1 1 1 1 1 1 1 ]
[ 4]  [ 3 3 1 1 1 1 ]
[ 5]  [ 3 3 3 1 ]
[ 6]  [ 4 2 2 2 ]
[ 7]  [ 4 4 2 ]
[ 8]  [ 5 1 1 1 1 1 ]
[ 9]  [ 5 3 1 1 ]
[10]  [ 5 5 ]
[11]  [ 6 2 2 ]
[12]  [ 6 4 ]
[13]  [ 7 1 1 1 ]
[14]  [ 7 3 ]
[15]  [ 8 2 ]
[16]  [ 9 1 ]
[17]  [ 10 ]
(End)

A. G. Elashvili and V. G. Kac, Classification of good gradings of simple Lie algebras. Lie groups and invariant theory, 85-104, Amer. Math. Soc. Transl. Ser. 2, 213, Amer. Math. Soc., Providence, RI, 2005.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Karin Baur and Nolan Wallach, Nice parabolic subalgebras of reductive Lie algebras, Represent. Theory 9 (2005), 1-29.
A. G. Elashvili and V. G. Kac, Classification of good gradings of simple Lie algebras, arXiv:math-ph/0312030, 2002-2004.
Sergi Elizalde and Emeric Deutsch, The degree of asymmetry of a sequence, Enum. Combinat. Applic. 2 (2022) no 1 #S2R7, U(0,z).

Bisections are A078408 and A096967.
The complement in partitions is counted by A006477
A version for compositions is A016116.
A pointed version is A035363, ranked by A066207.
A000041 counts integer partitions.
A025065 counts palindromic partitions.
A027187 counts partitions with even length/maximum.
A035377 counts partitions using multiples of 3.
A058696 counts partitions of even numbers, ranked by A300061.
A340785 counts factorizations into even factors.
Cf. A000009, A002865, A027383, A035457, A117298, A117989, A168021, A274230, A345170, A349060, A349061.

b:= proc(n, i) option remember; `if`(i>n, 0,
      `if`(irem(n, i)=0, 1, 0) +add(`if`(irem(j, 2)=0,
       b(n-i*j, i+1), 0), j=0..n/i))
    end:
a:= n-> b(n, 1):
seq(a(n), n=1..60);  # Alois P. Heinz, Mar 26 2014

(* The following Mathematica program first generates all of the palindromic, unimodal compositions of n and then counts them. *)
Pal[n_] := Block[{i, j, k, m, Q, L}, If[n == 1, Return[{{1}}]]; If[n == 2, Return[{{1, 1}, {2}}]]; L = {{n}}; If[Mod[n, 2] == 0, L = Append[L, {n/2, n/2}]]; For[i = 1, i < n, i++, Q = Pal[n - 2i]; m = Length[Q]; For[j = 1, j <= m, j++, If[i <= Q[[j, 1]], L = Append[L, Append[Prepend[Q[[j]], i], i]]]]]; L] NoPal[n_] := Length[Pal[n]]
a[n_] := PartitionsQ[n] + If[EvenQ[n], PartitionsP[n/2], 0]; Table[a[n], {n, 1, 55}] (* Jean-François Alcover, Mar 17 2014, after Vladeta Jovovic *)
Table[Length[Select[IntegerPartitions[n],And@@EvenQ/@Rest[Length/@Split[#]]&]],{n,1,30}] (* Gus Wiseman, Jan 13 2022 *)

my(x='x+O('x^66)); Vec(eta(x^2)/eta(x)+1/eta(x^2)-2) \\ Joerg Arndt, Jan 17 2016

G.f.: sum(j>=1, q^j * (1-q^j)/prod(i=1..j, 1-q^(2*i) ) ).
G.f.: F + G - 2, where F = Product_{j>=1} 1/(1-q^(2*j)), G = Product_{j>=0} 1/(1-q^(2*j+1)).
a(2*n) = A000041(n) + A000009(2*n); a(2*n-1) = A000009(2*n-1). - Vladeta Jovovic, Aug 11 2004
a(n) = A000009(n) + A035363(n) = A000041(n) - A006477(n). - Omar E. Pol, Aug 16 2013

A168021 Triangle T(n,k) read by rows in which row n lists the number of partitions of n into parts divisible by k.

Original entry on oeis.org

1, 2, 1, 3, 0, 1, 5, 2, 0, 1, 7, 0, 0, 0, 1, 11, 3, 2, 0, 0, 1, 15, 0, 0, 0, 0, 0, 1, 22, 5, 0, 2, 0, 0, 0, 1, 30, 0, 3, 0, 0, 0, 0, 0, 1, 42, 7, 0, 0, 2, 0, 0, 0, 0, 1, 56, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 77, 11, 5, 3, 0, 2, 0, 0, 0, 0, 0, 1

Offset: 1

Omar E. Pol, Nov 20 2009, Nov 21 2009

The row-reversed version is A168016.

Also see A168020.

			Triangle begins:
==============================================
...... k: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10 11 12
==============================================
n=1 ..... 1,
n=2 ..... 2, 1,
n=3 ..... 3, 0, 1,
n=4 ..... 5, 2, 0, 1,
n=5 ..... 7, 0, 0, 0, 1,
n=6 .... 11, 3, 2, 0, 0, 1,
n=7 .... 15, 0, 0, 0, 0, 0, 1,
n=8 .... 22, 5, 0, 2, 0, 0, 0, 1,
n=9 .... 30, 0, 3, 0, 0, 0, 0, 0, 1,
n=10 ... 42, 7, 0, 0, 2, 0, 0, 0, 0, 1,
n=11 ... 56, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
n=12 ... 77,11, 5, 3, 0, 2, 0, 0, 0, 0, 0, 1,
...

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
Omar E. Pol, Illustration of the shell model of partitions (2D and 3D)
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. A000007, A000012, A000041, A035377, A035444, A047968 (row sums).

Cf. A135010, A138121, A168016, A168017, A168018, A168019, A168020.

T[n_, k_]:= If[IntegerQ[n/k], PartitionsP[n/k], 0];
Table[T[n, k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Jan 12 2023 *)

def A168021(n,k): return number_of_partitions(n/k) if (n%k)==0 else 0
flatten([[A168021(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Jan 12 2023

T(n,k) = A000041(n/k) if k|n, else T(n,k)=0.
Sum_{k=1..n} T(n, k) = A047968(n).
From G. C. Greubel, Jan 12 2023: (Start)
T(2*n, n) = 2*A000012(n).
T(2*n-1, n+1) = A000007(n-2). (End)

Edited by Charles R Greathouse IV, Mar 23 2010

A168020 Square array read by antidiagonals in which row n lists the number of partitions of n into parts divisible by k.

Original entry on oeis.org

1, 2, 0, 3, 1, 0, 5, 0, 0, 0, 7, 2, 1, 0, 0, 11, 0, 0, 0, 0, 0, 15, 3, 0, 1, 0, 0, 0, 22, 0, 2, 0, 0, 0, 0, 0, 30, 5, 0, 0, 1, 0, 0, 0, 0, 42, 0, 0, 0, 0, 0, 0, 0, 0, 0, 56, 7, 3, 2, 0, 1, 0, 0, 0, 0, 0, 77, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 101, 11, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Omar E. Pol, Nov 20 2009

In the square array, note that the column k starts with k-1 zeros. Then list each partition number of positive integers followed by k-1 zeros. See A000041, which is the main entry for this sequence.

			The array, A(n, k), begins:
   n | k = 1   2   3   4   5   6   7   8   9  10  11  12
  ---+--------------------------------------------------
   1 |     1   0   0   0   0   0   0   0   0   0   0   0
   2 |     2   1   0   0   0   0   0   0   0   0   0   0
   3 |     3   0   1   0   0   0   0   0   0   0   0   0
   4 |     5   2   0   1   0   0   0   0   0   0   0   0
   5 |     7   0   0   0   1   0   0   0   0   0   0   0
   6 |    11   3   2   0   0   1   0   0   0   0   0   0
   7 |    15   0   0   0   0   0   1   0   0   0   0   0
   8 |    22   5   0   2   0   0   0   1   0   0   0   0
   9 |    30   0   3   0   0   0   0   0   1   0   0   0
  10 |    42   7   0   0   2   0   0   0   0   1   0   0
  11 |    56   0   0   0   0   0   0   0   0   0   1   0
  12 |    77  11   5   3   0   2   0   0   0   0   0   1
  ...
Antidiagonal triangle, T(n,k), begins as:
   1;
   2, 0;
   3, 1, 0;
   5, 0, 0, 0;
   7, 2, 1, 0, 0;
  11, 0, 0, 0, 0, 0;
  15, 3, 0, 1, 0, 0, 0;
  22, 0, 2, 0, 0, 0, 0, 0;
  30, 5, 0, 0, 1, 0, 0, 0, 0;
  42, 0, 0, 0, 0, 0, 0, 0, 0, 0;

G. C. Greubel, Antidiagonals n = 1..50, flattened
Omar E. Pol, Illustration of the shell model of partitions (2D and 3D)
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. A000007, A000041, A035377, A035444, A135010, A138121.

Cf. A168016, A168017, A168018, A168019, A168021. [From Omar E. Pol, Nov 23 2009]

T[n_, k_]:= If[IntegerQ[(n-k+1)/k], PartitionsP[(n-k+1)/k], 0];
Table[T[n, k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Jan 12 2023 *)

def A168020(n,k): return number_of_partitions((n-k+1)/k) if ((n-k+1)%k)==0 else 0
flatten([[A168020(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Jan 12 2023

A(n, k) = A000041(n/k) if k divides n, otherwise A(n, k) = 0 (array).
A(n, 1) = A(n*k, k) = A000041(n).
From G. C. Greubel, Jan 12 2023: (Start)
T(n, k) = A000041((n-k+1)/k) if k divides (n-k+1), otherwise T(n, k) = 0 (triangle).
T(n, 1) = A000041(n).
T(2*n, n) = 2*A000007(n-1), n >= 1. (End)

Edited by Omar E. Pol, Nov 21 2009
Edited by Charles R Greathouse IV, Mar 23 2010
Edited by Max Alekseyev, May 07 2010

A348550 Heinz numbers of integer partitions whose length is 2/3 their sum, rounded down.

Original entry on oeis.org

1, 3, 6, 9, 10, 18, 20, 36, 40, 54, 56, 60, 108, 112, 120, 216, 224, 240, 324, 336, 352, 360, 400, 648, 672, 704, 720, 800, 1296, 1344, 1408, 1440, 1600, 1664, 1944, 2016, 2112, 2160, 2240, 2400, 3328, 3888, 4032, 4224, 4320, 4480, 4800, 6656, 7776, 8064, 8448

Offset: 1

Gus Wiseman, Nov 05 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms and their prime indices begin:
    1: {}
    3: {2}
    6: {1,2}
    9: {2,2}
   10: {1,3}
   18: {1,2,2}
   20: {1,1,3}
   36: {1,1,2,2}
   40: {1,1,1,3}
   54: {1,2,2,2}
   56: {1,1,1,4}
   60: {1,1,2,3}
  108: {1,1,2,2,2}
  112: {1,1,1,1,4}
  120: {1,1,1,2,3}
  216: {1,1,1,2,2,2}
  224: {1,1,1,1,1,4}
  240: {1,1,1,1,2,3}

The partitions with these as Heinz numbers are counted by A108711.
An adjoint version is A347452, counted by A119620.
The unrounded version is A348384, counted by A035377.
A001222 counts prime factors with multiplicity.
A056239 adds up prime indices, row sums of A112798.
A316524 gives the alternating sum of prime indices, reverse A344616.
A344606 counts alternating permutations of prime factors.
Cf. A001105, A028982, A028260, A119899, A316413, A346703, A346704.

Select[Range[1000],Floor[2*Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]/3]==PrimeOmega[#]&]

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
isA348550(n) = (bigomega(n)==floor((2/3)*A056239(n))); \\ Antti Karttunen, Nov 08 2021

A001222(a(n)) = floor(2*A056239(a(n))/3).

A318026 Expansion of Product_{k>=1} 1/((1 - x^k)(1 - x^(3k))).

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 16, 22, 33, 50, 70, 98, 143, 193, 266, 368, 493, 659, 892, 1170, 1543, 2035, 2642, 3422, 4448, 5694, 7294, 9334, 11839, 14982, 18968, 23812, 29868, 37410, 46598, 57924, 71953, 88913, 109728, 135212, 165991, 203407, 248986, 303706, 369939, 449967, 545820, 661038, 799629

Offset: 0

Ilya Gutkovskiy, Aug 13 2018

nonn

Convolution of A000041 and A035377.
Convolution of A000712 and A137569.
Convolution inverse of A030203.
Number of partitions of n if there are 2 kinds of parts that are multiples of 3.

			a(4) = 6 because we have [4], [3, 1], [3', 1], [2, 2], [2, 1, 1] and [1, 1, 1, 1].

Zakir Ahmed, Nayandeep Deka Baruah, Manosij Ghosh Dastidar, New congruences modulo 5 for the number of 2-color partitions, Journal of Number Theory, Volume 157, December 2015, Pages 184-198.
Index entries for sequences related to partitions

Cf. A000041, A000712, A002513, A030203, A030206, A035377, A112175, A137569, A318027, A318028.

a:=series(mul(1/((1-x^k)*(1-x^(3*k))),k=1..55),x=0,49): seq(coeff(a,x,n),n=0..48); # Paolo P. Lava, Apr 02 2019

nmax = 48; CoefficientList[Series[Product[1/((1 - x^k) (1 - x^(3 k))), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[1/(QPochhammer[x] QPochhammer[x^3]), {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[Exp[Sum[x^k (1 + x^k + 2 x^(2 k))/(k (1 - x^(3 k))), {k, 1, nmax}]], {x, 0, nmax}], x]
Table[Sum[PartitionsP[k] PartitionsP[n - 3 k], {k, 0, n/3}], {n, 0, 48}]

G.f.: exp(Sum_{k>=1} x^k*(1 + x^k + 2*x^(2*k))/(k*(1 - x^(3*k)))).

a(n) ~ exp(2*sqrt(2*n)*Pi/3) / (3 * 2^(5/4) * n^(5/4)). - Vaclav Kotesovec, Aug 14 2018

A348384 Heinz numbers of integer partitions whose length is 2/3 their sum.

Original entry on oeis.org

1, 6, 36, 40, 216, 224, 240, 1296, 1344, 1408, 1440, 1600, 6656, 7776, 8064, 8448, 8640, 8960, 9600, 34816, 39936, 46656, 48384, 50176, 50688, 51840, 53760, 56320, 57600, 64000, 155648, 208896, 239616, 266240, 279936, 290304, 301056, 304128, 311040, 315392

Offset: 1

Gus Wiseman, Nov 13 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers whose sum of prime indices is 3/2 their number. Counting the partitions with these Heinz numbers gives A035377(n) = A000041(n/3) if n is a multiple of 3, otherwise 0.

			The terms and their prime indices begin:
     1: {}
     6: {1,2}
    36: {1,1,2,2}
    40: {1,1,1,3}
   216: {1,1,1,2,2,2}
   224: {1,1,1,1,1,4}
   240: {1,1,1,1,2,3}
  1296: {1,1,1,1,2,2,2,2}
  1344: {1,1,1,1,1,1,2,4}
  1408: {1,1,1,1,1,1,1,5}
  1440: {1,1,1,1,1,2,2,3}
  1600: {1,1,1,1,1,1,3,3}
  6656: {1,1,1,1,1,1,1,1,1,6}
  7776: {1,1,1,1,1,2,2,2,2,2}

David A. Corneth, Table of n, a(n) for n = 1..10000

These partitions are counted by A035377.
Rounding down gives A348550 or A347452, counted by A108711 or A119620.
A000041 counts integer partitions.
A001222 counts prime factors with multiplicity.
A056239 adds up prime indices, row sums of A112798.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A344606 counts alternating permutations of prime factors.
Cf. A000070, A000097, A028260, A028982, A032766, A236914, A316413, A347457, A348551.

Select[Range[1000],2*Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]==3*PrimeOmega[#]&]

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }
isA348384(n) = (A056239(n)==(3/2)*bigomega(n)); \\ Antti Karttunen, Nov 22 2021

The sequence contains n iff A056239(n) = 3*A001222(n)/2. Here, A056239 adds up prime indices, while A001222 counts them with multiplicity.

Intersection of A028260 and A347452.

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A035363 Number of partitions of n into even parts.

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A096441 Number of palindromic and unimodal compositions of n. Equivalently, the number of orbits under conjugation of even nilpotent n X n matrices.

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A168021 Triangle T(n,k) read by rows in which row n lists the number of partitions of n into parts divisible by k.

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A168020 Square array read by antidiagonals in which row n lists the number of partitions of n into parts divisible by k.

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A348550 Heinz numbers of integer partitions whose length is 2/3 their sum, rounded down.

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A318026 Expansion of Product_{k>=1} 1/((1 - x^k)*(1 - x^(3*k))).

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A348384 Heinz numbers of integer partitions whose length is 2/3 their sum.

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A318026 Expansion of Product_{k>=1} 1/((1 - x^k)(1 - x^(3k))).