A027336 -id:A027336 - cp's OEIS frontend

A364159 Number of integer partitions of n - 1 containing fewer 1's than any other part.

0, 1, 1, 2, 2, 3, 4, 5, 7, 9, 11, 15, 20, 23, 32, 40, 50, 61, 82, 95, 126, 149, 188, 228, 292, 337, 430, 510, 633, 748, 933, 1083, 1348, 1579, 1925, 2262, 2761, 3197, 3893, 4544, 5458, 6354, 7634, 8835, 10577, 12261, 14546, 16864, 19990, 23043, 27226, 31428

Offset: 0

Gus Wiseman, Jul 16 2023

nonn

Also integer partitions of n with least co-mode 1. Here, we define a co-mode in a multiset to be an element that appears at most as many times as each of the others. For example, the co-modes in {a,a,b,b,b,c,c} are {a,c}.

			The a(1) = 1 through a(8) = 7 partitions:
  (1)  (11)  (21)   (31)    (41)     (51)      (61)       (71)
             (111)  (1111)  (221)    (321)     (331)      (431)
                            (11111)  (2211)    (421)      (521)
                                     (111111)  (2221)     (3221)
                                               (1111111)  (3311)
                                                          (22211)
                                                          (11111111)

For mode instead of co-mode we have A241131, ranks A360015.
The case with only one 1 is A364062, ranks A364061.
Counts partitions ranked by A364158.
Counts positions of 1's in A364191, high A364192.
A362611 counts modes in prime factorization, triangle A362614.
A362613 counts co-modes in prime factorization, triangle A362615.
Ranking and counting partitions:
- A356862 = unique mode, counted by A362608
- A359178 = unique co-mode, counted by A362610
- A362605 = multiple modes, counted by A362607
- A362606 = multiple co-modes, counted by A362609
Cf. A027336, A124943, A237984, A327472, A363486, A363487.

Table[Length[Select[IntegerPartitions[n-1],Count[#,1]

A027337 Number of partitions of n that do not contain 3 as a part.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 8, 10, 15, 19, 27, 34, 47, 59, 79, 99, 130, 162, 209, 259, 330, 407, 512, 628, 783, 956, 1181, 1435, 1760, 2129, 2594, 3124, 3784, 4539, 5468, 6534, 7834, 9327, 11132, 13208, 15701, 18568, 21989, 25923, 30592, 35960, 42297, 49579, 58139, 67967

Offset: 0

Clark Kimberling

nonn

a(n) is also the number of partitions of n with less than three 1's. - Geoffrey Critzer, Jun 20 2014

Cf. A000041, A027336, A027338.
Column k=0 of A263232.
Column 3 of A175788.

nn=49;CoefficientList[Series[(1-x^3)Product[1/(1-x^i),{i,1,nn}],{x,0,nn}],x] (* Geoffrey Critzer, Jun 20 2014 *)

a(n)=if(n<0,0,polcoeff((1-x^3)/eta(x+x*O(x^n)),n))

G.f.: (1-x^3) Product_{m>0} 1/(1-x^m).
a(n) = A000041(n) - A000041(n-3).
a(n) ~ Pi * exp(sqrt(2*n/3)*Pi) / (4*sqrt(2)*n^(3/2)) * (1 - (3*sqrt(3/2)/Pi + Pi/(24*sqrt(6)) + 3*Pi/(2*sqrt(6)))/sqrt(n) + (37/8 + 9/(2*Pi^2) + 1801*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 04 2016

A363745 Number of integer partitions of n whose rounded-down mean is 2.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 3, 4, 10, 6, 16, 21, 24, 32, 58, 47, 85, 111, 119, 158, 248, 217, 341, 442, 461, 596, 867, 792, 1151, 1465, 1506, 1916, 2652, 2477, 3423, 4298, 4381, 5488, 7334, 6956, 9280, 11503, 11663, 14429, 18781, 17992, 23383, 28675, 28970, 35449, 45203

Offset: 0

Gus Wiseman, Jul 05 2023

nonn

			The a(2) = 1 through a(10) = 16 partitions:
  (2)  .  (22)  (32)  (222)  (322)  (332)   (3222)  (3322)
          (31)  (41)  (321)  (331)  (422)   (3321)  (3331)
                      (411)  (421)  (431)   (4221)  (4222)
                             (511)  (521)   (4311)  (4321)
                                    (611)   (5211)  (4411)
                                    (2222)  (6111)  (5221)
                                    (3221)          (5311)
                                    (3311)          (6211)
                                    (4211)          (7111)
                                    (5111)          (22222)
                                                    (32221)
                                                    (33211)
                                                    (42211)
                                                    (43111)
                                                    (52111)
                                                    (61111)

For 1 instead of 2 we have A025065, ranks A363949.
The high version is A026905 reduplicated, ranks A363950.
Column k = 2 of A363945.
These partitions have ranks A363954.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean, median A000975.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
A349156 counts partitions with non-integer mean, ranks A348551.
Cf. A000041, A002865, A027336, A237984, A241131, A327472, A327482, A363723, A363943, A363944, A363946.

Table[Length[Select[IntegerPartitions[n],Floor[Mean[#]]==2&]],{n,0,30}]

A104384 Number of partitions of triangular numbers n*(n+1)/2 into (n-2) distinct parts for n>=3.

Original entry on oeis.org

1, 4, 12, 27, 57, 110, 201, 352, 598, 984, 1586, 2503, 3882, 5928, 8932, 13287, 19551, 28472, 41078, 58754, 83372, 117417, 164230, 228212, 315190, 432817, 591130, 803192, 1086035, 1461680, 1958596, 2613417, 3473190, 4598073, 6064920, 7971480

Offset: 3

Paul D. Hanna, Mar 04 2005

nonn

In triangle A104382, equals the second diagonal down from the main diagonal.

Also equals a diagonal with slope -3 in the Partition Numbers triangle A008284, found at n = 3+3k, or T(3+3k,k) for k >=1. - Richard R. Forberg, Dec 02 2014

Vaclav Kotesovec, Table of n, a(n) for n = 3..132

Cf. A000009, A104382, A008284.

{a(n)=if(n<3,0,polcoeff(polcoeff( prod(i=1,n*(n+1)/2,1+y*x^i,1+x*O(x^(n*(n+1)/2))),n*(n+1)/2,x),n-2,y))}

From Álvar Ibeas, Jul 23 2020: (Start)
Writing p(m) for A000041(m),
a(n) = p(2n-1) - A000070(n) + 1 and
a(n+1) - a(n) = p(2*n+1) - p(2*n-1) - p(n+1) = A027336(2*n+1) - p(n+1).
(End)

A174455 Number of partitions where the number of 1's and 2's are equal.

Original entry on oeis.org

1, 0, 0, 2, 1, 1, 4, 3, 4, 8, 8, 10, 17, 18, 23, 34, 39, 48, 67, 78, 97, 127, 151, 185, 237, 281, 343, 428, 511, 616, 759, 902, 1084, 1315, 1562, 1863, 2242, 2649, 3147, 3752, 4424, 5222, 6190, 7266, 8545, 10062, 11776, 13782, 16157, 18832, 21964, 25622, 29777, 34589, 40200, 46556, 53912

Offset: 0

Joerg Arndt, Nov 28 2010

nonn

From Omar E. Pol, Jan 19 2013: (Start)
Column 3 of triangle A220504.
With offset 3, a(n) is also the number of appearances of 3 as the smallest part in all partitions of n.
Also consider the sequence formed by [0, 0] together with this sequence, with offset 1, then it appears that A027336(n) = Sum_{j=1..3} a(n+j), n >= 0.
(End)

			a(8)=9, there are 8 such partitions of 9, they are
  #1:    9 =  3* 1 + 3* 2 + 0    + 0    + 0    + 0    + 0    + 0    + 0
  #2:    9 =  2* 1 + 2* 2 + 1* 3 + 0    + 0    + 0    + 0    + 0    + 0
  #3:    9 =  1* 1 + 1* 2 + 2* 3 + 0    + 0    + 0    + 0    + 0    + 0
  #4:    9 =  0    + 0    + 3* 3 + 0    + 0    + 0    + 0    + 0    + 0
  #5:    9 =  0    + 0    + 0    + 1* 4 + 1* 5 + 0    + 0    + 0    + 0
  #6:    9 =  1* 1 + 1* 2 + 0    + 0    + 0    + 1* 6 + 0    + 0    + 0
  #7:    9 =  0    + 0    + 1* 3 + 0    + 0    + 1* 6 + 0    + 0    + 0
  #8:    9 =  0    + 0    + 0    + 0    + 0    + 0    + 0    + 0    + 1* 9

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Cf. A008483, A182713, A240056.

b:= proc(n, i) option remember; local j, r; if n=0 or i<1 then 0
      else `if`(i=3 and irem(n, 3, 'r')=0, r, 0); for j from 0
      to n/i do %+b(n-i*j, i-1) od; % fi
    end:
a:= n-> b(n+3, n+3):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 20 2013

(* See A240056. - Clark Kimberling, Mar 31 2014 *)
m = 66; gf = 1/((1-x^3)*Product[1-x^n, {n, 3, m}]) + O[x]^m; CoefficientList[gf, x] (* Jean-François Alcover, Jul 02 2015, after Joerg Arndt *)

N=66;  x='x+O('x^N);
gf=1/( (1-x^3) * prod(n=3,N, 1-x^n) );
Vec(gf)
/* Joerg Arndt, Jul 07 2012 */

G.f.: 1/( (1-x^3) * Product_{n>=3} (1-x^n) ). - Joerg Arndt, Jul 07 2012
a(n) = A182713(n+2) - A182713(n) = A240056(n+1) - A240056(n) for n >= 0. - Clark Kimberling, Mar 31 2014
a(n) ~ Pi * exp(sqrt(2*n/3)*Pi) / (9 * 2^(3/2) * n^(3/2)). - Vaclav Kotesovec, Jan 15 2022

A364056 Numbers whose prime factors have high median 2. Numbers whose prime factors (with multiplicity) are mostly 2's.

Original entry on oeis.org

2, 4, 8, 12, 16, 20, 24, 28, 32, 40, 44, 48, 52, 56, 64, 68, 72, 76, 80, 88, 92, 96, 104, 112, 116, 120, 124, 128, 136, 144, 148, 152, 160, 164, 168, 172, 176, 184, 188, 192, 200, 208, 212, 224, 232, 236, 240, 244, 248, 256, 264, 268, 272, 280, 284, 288, 292

Offset: 1

Gus Wiseman, Jul 07 2023

nonn

The multiset of prime factors of n is row n of A027746.

The high median (see A124944) in a multiset is either the middle part (for odd length), or the greatest of the two middle parts (for even length).

			The terms together with their prime indices begin:
     2: {1}             64: {1,1,1,1,1,1}      136: {1,1,1,7}
     4: {1,1}           68: {1,1,7}            144: {1,1,1,1,2,2}
     8: {1,1,1}         72: {1,1,1,2,2}        148: {1,1,12}
    12: {1,1,2}         76: {1,1,8}            152: {1,1,1,8}
    16: {1,1,1,1}       80: {1,1,1,1,3}        160: {1,1,1,1,1,3}
    20: {1,1,3}         88: {1,1,1,5}          164: {1,1,13}
    24: {1,1,1,2}       92: {1,1,9}            168: {1,1,1,2,4}
    28: {1,1,4}         96: {1,1,1,1,1,2}      172: {1,1,14}
    32: {1,1,1,1,1}    104: {1,1,1,6}          176: {1,1,1,1,5}
    40: {1,1,1,3}      112: {1,1,1,1,4}        184: {1,1,1,9}
    44: {1,1,5}        116: {1,1,10}           188: {1,1,15}
    48: {1,1,1,1,2}    120: {1,1,1,2,3}        192: {1,1,1,1,1,1,2}
    52: {1,1,6}        124: {1,1,11}           200: {1,1,1,3,3}
    56: {1,1,1,4}      128: {1,1,1,1,1,1,1}    208: {1,1,1,1,6}

Partitions of this type are counted by A027336.
Median of prime indices is A360005(n)/2.
For mode instead of median we have A360013, low A360015.
The low version is A363488, positions of 1's in A363941.
Positions of 1's in A363942.
A112798 lists prime indices, length A001222, sum A056239.
A123528/A123529 gives mean of prime factors, indices A326567/A326568.
A124943 counts partitions by low median, high A124944.
Cf. A072978, A215366, A316413, A359908, A363727, A363740, A363949.

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
merr[y_]:=If[Length[y]==0,0,If[OddQ[Length[y]],y[[(Length[y]+1)/2]], y[[1+Length[y]/2]]]];
Select[Range[100],merr[prifacs[#]]==2&]

A027343 Number of partitions of n that do not contain 9 as a part.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 41, 54, 74, 96, 128, 165, 216, 275, 355, 448, 571, 715, 901, 1120, 1399, 1727, 2139, 2625, 3228, 3938, 4812, 5840, 7094, 8568, 10352, 12447, 14967, 17919, 21450, 25581, 30496, 36234, 43031, 50951, 60292

Offset: 0

Clark Kimberling

nonn

9th column of A175788. Cf. A000041, A027336, A027337-A027344.

A41:= n-> `if`(n<0, 0, combinat[numbpart](n)):
a:= n-> A41(n) -A41(n-9):
seq(a(n), n=0..50);

Table[Count[IntegerPartitions[n],?(FreeQ[#,9]&)],{n,0,50}] (* _Harvey P. Dale, Oct 02 2022 *)

G.f.: (1-x^9) Product_{m>0} 1/(1-x^m).
a(n) = A000041(n)-A000041(n-9).
a(n) ~ 3*Pi * exp(sqrt(2*n/3)*Pi) / (4*sqrt(2)*n^(3/2)) * (1 - (3*sqrt(3/2)/Pi + Pi/(24*sqrt(6)) + 9*Pi/(2*sqrt(6)))/sqrt(n) + (109/8 + 9/(2*Pi^2) + 15769*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 04 2016

Edited by Alois P. Heinz, Dec 04 2010

A027344 Number of partitions of n that do not contain 10 as a part.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 55, 75, 98, 130, 169, 220, 282, 363, 460, 585, 736, 925, 1154, 1440, 1782, 2205, 2713, 3333, 4075, 4977, 6050, 7347, 8888, 10735, 12925, 15541, 18627, 22297, 26620, 31734, 37741, 44825, 53118, 62865

Offset: 0

Clark Kimberling

nonn

10th column of A175788. Cf. A000041, A027336, A027337-A027343.

A41:= n-> `if`(n<0, 0, combinat[numbpart](n)):
a:= n-> A41(n) -A41(n-10):
seq(a(n), n=0..50);

G.f.: (1-x^10) Product_{m>0} 1/(1-x^m).
a(n) = A000041(n)-A000041(n-10).
a(n) ~ 5*Pi * exp(sqrt(2*n/3)*Pi) / (6*sqrt(2)*n^(3/2)) * (1 - (3*sqrt(3/2)/Pi + Pi/(24*sqrt(6)) + 10*Pi/(2*sqrt(6)))/sqrt(n) + (121/8 + 9/(2*Pi^2) + 19441*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 04 2016

More terms from Benoit Cloitre, Dec 10 2002

Edited by Alois P. Heinz, Dec 04 2010

A121659 Number of partitions of n into parts with at most one part not greater than 2.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 9, 12, 15, 20, 25, 32, 40, 51, 63, 79, 97, 121, 148, 182, 221, 271, 328, 398, 479, 579, 694, 834, 995, 1190, 1415, 1684, 1995, 2366, 2793, 3298, 3881, 4569, 5360, 6288, 7355, 8603, 10037, 11705, 13619, 15842, 18388, 21333, 24703

Offset: 1

Reinhard Zumkeller, Aug 14 2006

nonn

The number of partitions of n that have the first part at least thrice larger than the second part. Example: a(8) = #{8, 7+1, 6+2, 6+1+1, 5+1+1+1, 4+1+1+1+1, 3+1+1+1+1+1}=7. [Mircea Merca, Jul 24 2011]

			a(8) = #{8,7+1,6+2,5+3,4+4,4+3+1,3+3+2} = 7;
a(9) = #{9,8+1,7+2,6+3,5+4,5+3+1,4+4+1,4+3+2,3+3+3} = 9.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Mircea Merca, Fast algorithm for generating ascending compositions, arXiv:1903.10797 [math.CO], 2019.

Cf. A027336, A008483, A121081, A000041.

a(n) = A121081(n) - A008483(n-3) for n>2.
a(n) = p(n) - p(n-2) - p(n-3) + p(n-5) where p(n) = A000041(n). See Merca p. 6. - Michel Marcus, Mar 27 2019
a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi^2 / (4*sqrt(3)*n^2) * (1 - (18/Pi + 61*Pi/24)/sqrt(6*n)). - Vaclav Kotesovec, Jan 15 2022

A173301 a(n) = A000041(2^n - 1).

Original entry on oeis.org

1, 1, 3, 15, 176, 6842, 1505499, 3913864295, 338854264248680, 4216199393504640098482, 59475094770587936660132803278445, 17618334934720173062514849536736413843694654543

Offset: 0

Gary W. Adamson, Feb 15 2010

nonn

The partition numbers have an apparent fractal-like structure starting with every term in A173301.
Let A000041 = row 0, then under every (2^n - 1)-th term, begin a new row with the partition numbers; then take finite differences of each column from below.
The sum of finite difference terms will reproduce the partition numbers, with finite difference rows (starting from the top going down) = number of partitions of n that do not contain (1, 2, 3,...). (Cf. the array shown in A173302).

Refer to tables of the partition numbers.

Amiram Eldar, Table of n, a(n) for n = 0..19

Cf. A000041, A000225, A002865, A027336, A173302.

Table[PartitionsP[2^n - 1], {n, 0 ,10}] (* Amiram Eldar, Feb 26 2020 *)

a(n) = A000041(2^n - 1), n = (0, 1, 2,...).

a(n) = A000041(A000225(n)). - Omar E. Pol, Oct 29 2013

cp's OEIS Frontend

A364159 Number of integer partitions of n - 1 containing fewer 1's than any other part.

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A027337 Number of partitions of n that do not contain 3 as a part.

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A363745 Number of integer partitions of n whose rounded-down mean is 2.

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A104384 Number of partitions of triangular numbers n*(n+1)/2 into (n-2) distinct parts for n>=3.

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A174455 Number of partitions where the number of 1's and 2's are equal.

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A364056 Numbers whose prime factors have high median 2. Numbers whose prime factors (with multiplicity) are mostly 2's.

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A027343 Number of partitions of n that do not contain 9 as a part.

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A027344 Number of partitions of n that do not contain 10 as a part.

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Extensions

A121659 Number of partitions of n into parts with at most one part not greater than 2.