cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A237753 Number of partitions of n such that 2*(greatest part) = (number of parts).

Original entry on oeis.org

0, 1, 0, 0, 1, 1, 1, 2, 1, 2, 3, 4, 5, 7, 7, 9, 12, 15, 17, 23, 27, 34, 42, 50, 60, 75, 87, 106, 128, 154, 182, 222, 260, 311, 369, 437, 515, 613, 716, 845, 993, 1166, 1361, 1599, 1861, 2176, 2534, 2950, 3422, 3983, 4605, 5339, 6174, 7136, 8227, 9500, 10928
Offset: 1

Views

Author

Clark Kimberling, Feb 13 2014

Keywords

Comments

Also, the number of partitions of n such that (greatest part) = 2*(number of parts); hence, the number of partitions of n such that (rank + greatest part) = 0.

Examples

			a(8) = 2 counts these partitions:  311111, 2222.

Links

Crossrefs

Column 2 of A350879.
Cf. A064173, A237751, A237752, A237754-A237757, A350893.

Programs

  • Mathematica
    z = 50; Table[Count[IntegerPartitions[n], p_ /; 2 Max[p] = = Length[p]], {n, z}]
(* or *)
nmax = 100; Rest[CoefficientList[Series[Sum[x^(3*k-1) * Product[(1 - x^(2*k+j-1)) / (1 - x^j), {j, 1, k-1}], {k, 1, nmax/3 + 1}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Oct 15 2024 *)
nmax = 100; p = x; s = x; Do[p = Normal[Series[p*x^3*(1 - x^(3*k - 1))*(1 - x^(3*k))*(1 - x^(3*k + 1))/((1 - x^(2*k + 1))*(1 - x^(2*k))*(1 - x^k)), {x, 0, nmax}]]; s += p;, {k, 1, nmax/3 + 1}]; Take[CoefficientList[s, x], nmax] (* Vaclav Kotesovec, Oct 16 2024 *)
  • PARI
    my(N=66, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(3*k-1)*prod(j=1, k-1, (1-x^(2*k+j-1))/(1-x^j))))) \\ Seiichi Manyama, Jan 24 2022

Formula

G.f.: Sum_{k>=1} x^(3*k-1) * Product_{j=1..k-1} (1-x^(2*k+j-1))/(1-x^j). - Seiichi Manyama, Jan 24 2022
a(n) ~ Pi^2 * exp(Pi*sqrt(2*n/3)) / (4 * 3^(3/2) * n^2). - Vaclav Kotesovec, Oct 17 2024

A237755 Number of partitions of n such that 2*(greatest part) >= (number of parts).

Original entry on oeis.org

1, 2, 2, 4, 6, 9, 12, 18, 24, 34, 46, 63, 83, 111, 144, 190, 245, 318, 405, 520, 657, 833, 1045, 1312, 1634, 2036, 2517, 3114, 3829, 4705, 5751, 7027, 8544, 10381, 12564, 15190, 18301, 22026, 26425, 31669, 37849, 45180, 53796, 63983, 75923, 89987, 106435
Offset: 1

Views

Author

Clark Kimberling, Feb 13 2014

Keywords

Comments

Also, the number of partitions of n such that (greatest part) <= 2*(number of parts); hence, the number of partitions of n such that (rank + greatest part) >= 0.

Examples

			a(6) = 9 counts all of the 11 partitions of 6 except these:  21111, 111111.

Links

Crossrefs

Cf. A064173, A237751-A237755, A237756, A237757, A000041.

Programs

  • Mathematica
    z = 50; Table[Count[IntegerPartitions[n], p_ /; 2 Max[p] >= Length[p]], {n, z}]
  • PARI
    {a(n) = my(A); A = sum(m=0,n,x^m*prod(k=1,m,(1-x^(2*m+k-1))/(1-x^k +x*O(x^n)))); polcoeff(A,n)}
for(n=1,60,print1(a(n),", ")) \\ Paul D. Hanna, Aug 03 2015

Formula

a(n) = A000041(n) - A237751(n).
G.f.: Sum_{n>=1} x^n * Product_{k=1..n} (1 - x^(2*n+k-1))/(1 - x^k). - Paul D. Hanna, Aug 03 2015

A361858 Number of integer partitions of n such that the maximum is less than twice the median.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 8, 12, 15, 19, 22, 31, 34, 45, 55, 67, 78, 100, 115, 144, 170, 203, 238, 291, 337, 403, 473, 560, 650, 772, 889, 1046, 1213, 1414, 1635, 1906, 2186, 2533, 2913, 3361, 3847, 4433, 5060, 5808, 6628, 7572, 8615, 9835, 11158, 12698, 14394
Offset: 1

Views

Author

Gus Wiseman, Apr 02 2023

Keywords

Comments

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Examples

			The a(1) = 1 through a(8) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (11111)  (222)     (322)      (71)
                                     (321)     (331)      (332)
                                     (2211)    (2221)     (431)
                                     (111111)  (1111111)  (2222)
                                                          (3221)
                                                          (3311)
                                                          (22211)
                                                          (11111111)
The partition y = (3,2,2,1) has maximum 3 and median 2, and 3 < 2*2, so y is counted under a(8).

Crossrefs

For minimum instead of median we have A053263.
For length instead of median we have A237754.
Allowing equality gives A361848, strict A361850.
The equal version is A361849, ranks A361856.
For mean instead of median we have A361852.
Reversing the inequality gives A361857, ranks A361867.
The complement is counted by A361859, ranks A361868.
A000041 counts integer partitions, strict A000009.
A000975 counts subsets with integer median.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median.
A360005 gives twice median of prime indices, distinct A360457.
Cf. A008284, A027193, A237751, A237755, A237820, A237824, A240219, A361394, A361851, A361860, A361907.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Max@@#<2*Median[#]&]],{n,30}]

A361907 Number of integer partitions of n such that (length) * (maximum) > 2*n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 3, 4, 7, 11, 19, 26, 43, 60, 80, 115, 171, 201, 297, 374, 485, 656, 853, 1064, 1343, 1758, 2218, 2673, 3477, 4218, 5423, 6523, 7962, 10017, 12104, 14409, 17978, 22031, 26318, 31453, 38176, 45442, 55137, 65775, 77451, 92533, 111485, 131057
Offset: 1

Views

Author

Gus Wiseman, Mar 29 2023

Keywords

Comments

Also partitions such that (maximum) > 2*(mean).
These are partitions whose complement (see example) has size > n.

Examples

			The a(7) = 3 through a(10) = 11 partitions:
  (511)    (611)     (711)      (721)
  (4111)   (5111)    (5211)     (811)
  (31111)  (41111)   (6111)     (6211)
           (311111)  (42111)    (7111)
                     (51111)    (52111)
                     (411111)   (61111)
                     (3111111)  (421111)
                                (511111)
                                (3211111)
                                (4111111)
                                (31111111)
The partition y = (3,2,1,1) has length 4 and maximum 3, and 4*3 is not > 2*7, so y is not counted under a(7).
The partition y = (4,2,1,1) has length 4 and maximum 4, and 4*4 is not > 2*8, so y is not counted under a(8).
The partition y = (5,1,1,1) has length 4 and maximum 5, and 4*5 > 2*8, so y is counted under a(8).
The partition y = (5,2,1,1) has length 4 and maximum 5, and 4*5 > 2*9, so y is counted under a(9).
The partition y = (3,2,1,1) has diagram:
  o o o
  o o .
  o . .
  o . .
with complement (shown in dots) of size 5, and 5 is not > 7, so y is not counted under a(7).

Crossrefs

For length instead of mean we have A237751, reverse A237754.
For minimum instead of mean we have A237820, reverse A053263.
The complement is counted by A361851, median A361848.
Reversing the inequality gives A361852.
The equal version is A361853.
For median instead of mean we have A361857, reverse A361858.
Allowing equality gives A361906, median A361859.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A051293 counts subsets with integer mean.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
A116608 counts partitions by number of distinct parts.
A268192 counts partitions by complement size, ranks A326844.
Cf. A027193, A111907, A237752, A237755, A237821, A237824, A237984, A324562, A326622, A327482, A349156, A360071, A361394.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Length[#]*Max@@#>2n&]],{n,30}]

A237752 Number of partitions of n such that 2*(greatest part) <= (number of parts).

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 4, 6, 7, 10, 13, 18, 23, 31, 39, 50, 64, 82, 102, 130, 162, 203, 252, 313, 384, 475, 580, 710, 864, 1053, 1273, 1544, 1859, 2240, 2688, 3224, 3851, 4602, 5476, 6514, 7727, 9160, 10826, 12791, 15072, 17747, 20853, 24481, 28679, 33577, 39231
Offset: 1

Views

Author

Clark Kimberling, Feb 13 2014

Keywords

Comments

Also, the number of partitions of n such that (greatest part) >= 2*(number of parts); hence, the number of partitions of n such that (rank + greatest part) <= 0.
Also, the number of partitions p of n such that max(max(p), 2*(number of parts of p)) is a part of p.

Examples

			The partitions of 6 that do not qualify are 22311, 21111, 111111, so that a(6) = 11 - 3 = 8.

Links

Crossrefs

Cf. A064173, A237751, A237753-A237757, A000041.

Programs

  • Mathematica
    z = 50; Table[Count[IntegerPartitions[n], p_ /; 2 Max[p] <= Length[p]], {n, z}]
(* also *)
Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Max[Max[p],2*Length[p]]]], {n, 50}]

Formula

a(n) = A000041(n) - A237754(n).

A361857 Number of integer partitions of n such that the maximum is greater than twice the median.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 7, 11, 16, 25, 37, 52, 74, 101, 138, 185, 248, 325, 428, 554, 713, 914, 1167, 1476, 1865, 2336, 2922, 3633, 4508, 5562, 6854, 8405, 10284, 12536, 15253, 18489, 22376, 26994, 32507, 39038, 46802, 55963, 66817, 79582, 94643, 112315
Offset: 1

Views

Author

Gus Wiseman, Apr 02 2023

Keywords

Comments

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Examples

			The a(5) = 1 through a(10) = 16 partitions:
  (311)  (411)   (511)    (521)     (522)      (622)
         (3111)  (4111)   (611)     (621)      (721)
                 (31111)  (4211)    (711)      (811)
                          (5111)    (5211)     (5221)
                          (32111)   (6111)     (5311)
                          (41111)   (33111)    (6211)
                          (311111)  (42111)    (7111)
                                    (51111)    (43111)
                                    (321111)   (52111)
                                    (411111)   (61111)
                                    (3111111)  (331111)
                                               (421111)
                                               (511111)
                                               (3211111)
                                               (4111111)
                                               (31111111)
The partition y = (5,2,2,1) has maximum 5 and median 2, and 5 > 2*2, so y is counted under a(10).

Crossrefs

For length instead of median we have A237751.
For minimum instead of median we have A237820.
The complement is counted by A361848.
The equal version is A361849, ranks A361856.
Reversing the inequality gives A361858.
Allowing equality gives A361859, ranks A361868.
These partitions have ranks A361867.
For mean instead of median we have A361907.
A000041 counts integer partitions, strict A000009.
A000975 counts subsets with integer median.
A325347 counts partitions with integer median, complement A307683.
A359893 and A359901 count partitions by median.
A360005 gives twice median of prime indices, distinct A360457.
Cf. A008284, A013580, A027193, A061395, A237755, A237824, A240219, A361394, A361851, A361860.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Max@@#>2*Median[#]&]],{n,30}]

A361867 Positive integers > 1 whose prime indices satisfy (maximum) > 2*(median).

Original entry on oeis.org

20, 28, 40, 44, 52, 56, 66, 68, 76, 78, 80, 84, 88, 92, 99, 102, 104, 112, 114, 116, 117, 120, 124, 132, 136, 138, 148, 152, 153, 156, 160, 164, 168, 170, 171, 172, 174, 176, 184, 186, 188, 190, 198, 200, 204, 207, 208, 212, 220, 222, 224, 228, 230, 232, 234
Offset: 1

Views

Author

Gus Wiseman, Apr 05 2023

Keywords

Comments

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 median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

Examples

			The prime indices of 84 are {1,1,2,4}, with maximum 4 and median 3/2, and 4 > 2*(3/2), so 84 is in the sequence.
The terms together with their prime indices begin:
   20: {1,1,3}
   28: {1,1,4}
   40: {1,1,1,3}
   44: {1,1,5}
   52: {1,1,6}
   56: {1,1,1,4}
   66: {1,2,5}
   68: {1,1,7}
   76: {1,1,8}
   78: {1,2,6}
   80: {1,1,1,1,3}
   84: {1,1,2,4}
   88: {1,1,1,5}
   92: {1,1,9}
   99: {2,2,5}

Crossrefs

The LHS is A061395 (greatest prime index).
The RHS is A360005 (twice median), distinct A360457.
The equal version is A361856, counted by A361849.
These partitions are counted by A361857, reverse A361858.
Including the equal case gives A361868, counted by A361859.
For mean instead of median we have A361907.
A000975 counts subsets with integer median.
A001222 counts prime factors, distinct A001221.
A112798 lists prime indices, sum A056239.
Cf. A053263, A067801, A111907, A237751, A237820, A359893, A361848, A361855, A361908, A361909.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100], Max@@prix[#]>2*Median[prix[#]]&]

A237754 Number of partitions of n such that 2*(greatest part) > (number of parts).

Original entry on oeis.org

1, 1, 2, 4, 5, 8, 11, 16, 23, 32, 43, 59, 78, 104, 137, 181, 233, 303, 388, 497, 630, 799, 1003, 1262, 1574, 1961, 2430, 3008, 3701, 4551, 5569, 6805, 8284, 10070, 12195, 14753, 17786, 21413, 25709, 30824, 36856, 44014, 52435, 62384, 74062, 87811, 103901
Offset: 1

Views

Author

Clark Kimberling, Feb 13 2014

Keywords

Comments

Also, the number of partitions of n such that (greatest part) < 2*(number of parts); hence, the number of partitions of n such that (rank + greatest part) > 0.
Also, the number of partitions p of n such that max(max(p), 2*(number of parts of p)) is not a part of p.

Examples

			a(5) = 5 counts these partitions:  5, 41, 32, 311, 221.

Links

Crossrefs

Cf. A064173, A237751-A237753, A237755-A237757, A000041.

Programs

  • Mathematica
    z = 50; Table[Count[IntegerPartitions[n], p_ /; 2 Max[p] > Length[p]], {n, z}]
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, x^k*prod(j=1, k, (1-x^(2*k+j-2))/(1-x^j)))) \\ Seiichi Manyama, Jan 25 2022

Formula

a(n) = A000041(n) - A237752(n).
G.f.: Sum_{k>=1} x^k * Product_{j=1..k} (1-x^(2*k+j-2))/(1-x^j). - Seiichi Manyama, Jan 25 2022
Showing 1-8 of 8 results.

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