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

A361859 Number of integer partitions of n such that the maximum is greater than or equal to twice the median.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 7, 10, 15, 23, 34, 46, 67, 90, 121, 164, 219, 285, 375, 483, 622, 799, 1017, 1284, 1621, 2033, 2537, 3158, 3915, 4832, 5953, 7303, 8930, 10896, 13248, 16071, 19451, 23482, 28272, 33977, 40736, 48741, 58201, 69367, 82506, 97986, 116139
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(4) = 1 through a(9) = 15 partitions:
  (211)  (311)   (411)    (421)     (422)      (522)
         (2111)  (3111)   (511)     (521)      (621)
                 (21111)  (3211)    (611)      (711)
                          (4111)    (4211)     (4221)
                          (22111)   (5111)     (4311)
                          (31111)   (32111)    (5211)
                          (211111)  (41111)    (6111)
                                    (221111)   (33111)
                                    (311111)   (42111)
                                    (2111111)  (51111)
                                               (321111)
                                               (411111)
                                               (2211111)
                                               (3111111)
                                               (21111111)
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 A237752.
For minimum instead of median we have A237821.
Reversing the inequality gives A361848.
The equal case is A361849, ranks A361856.
The unequal case is A361857, ranks A361867.
The complement is counted by A361858.
These partitions have ranks A361868.
For mean instead of median we have A361906.
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, A067538, A237755, A237820, A237824, A240219, A359907, A361851, A361860, A361907.

Programs

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

A361906 Number of integer partitions of n such that (length) * (maximum) >= 2*n.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 3, 5, 9, 15, 19, 36, 43, 68, 96, 125, 171, 232, 297, 418, 529, 676, 853, 1156, 1393, 1786, 2316, 2827, 3477, 4484, 5423, 6677, 8156, 10065, 12538, 15121, 17978, 22091, 26666, 32363, 38176, 46640, 55137, 66895, 79589, 92621, 111485, 133485
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(6) = 2 through a(10) = 15 partitions:
  (411)   (511)    (611)     (621)      (721)
  (3111)  (4111)   (4211)    (711)      (811)
          (31111)  (5111)    (5211)     (5221)
                   (41111)   (6111)     (5311)
                   (311111)  (42111)    (6211)
                             (51111)    (7111)
                             (321111)   (42211)
                             (411111)   (43111)
                             (3111111)  (52111)
                                        (61111)
                                        (421111)
                                        (511111)
                                        (3211111)
                                        (4111111)
                                        (31111111)
The partition y = (4,2,1,1) has length 4 and maximum 4, and 4*4 >= 2*8, so y is counted under a(8).
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 = (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 A237752, reverse A237755.
For minimum instead of mean we have A237821, reverse A237824.
For median instead of mean we have A361859, reverse A361848.
The unequal case is A361907.
The complement is counted by A361852.
The equal case is A361853, ranks A361855.
Reversing the inequality gives A361851.
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.
A268192 counts partitions by complement size, ranks A326844.
Cf. A027193, A111907, A116608, A237984, A324521, A327482, A349156, A360068.

Programs

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

A237751 Number of partitions of n such that 2*(greatest part) < (number of parts).

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 3, 4, 6, 8, 10, 14, 18, 24, 32, 41, 52, 67, 85, 107, 135, 169, 210, 263, 324, 400, 493, 604, 736, 899, 1091, 1322, 1599, 1929, 2319, 2787, 3336, 3989, 4760, 5669, 6734, 7994, 9465, 11192, 13211, 15571, 18319, 21531, 25257, 29594, 34626
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) = 2 counts these partitions:  21111, 111111.

Links

Crossrefs

Cf. A064173, A237752-A237757, A000041.

Programs

  • Mathematica
    z = 55; Table[Count[IntegerPartitions[n], p_ /; 2 Max[p] < Length[p]], {n, z}]

Formula

a(n) = A000041(n) - A237755(n).

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

A361909 Positive integers > 1 whose prime indices satisfy: (maximum) = 2*(length).

Original entry on oeis.org

3, 14, 21, 35, 49, 52, 78, 117, 130, 152, 182, 195, 228, 273, 286, 325, 338, 342, 380, 429, 455, 464, 507, 513, 532, 570, 637, 696, 715, 798, 836, 845, 855, 950, 988, 1001, 1044, 1160, 1183, 1184, 1197, 1254, 1292, 1330, 1425, 1444, 1482, 1566, 1573, 1624
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.

Examples

			The terms together with their prime indices begin:
     3: {2}
    14: {1,4}
    21: {2,4}
    35: {3,4}
    49: {4,4}
    52: {1,1,6}
    78: {1,2,6}
   117: {2,2,6}
   130: {1,3,6}
   152: {1,1,1,8}
   182: {1,4,6}
   195: {2,3,6}
   228: {1,1,2,8}
   273: {2,4,6}
   286: {1,5,6}
   325: {3,3,6}
   338: {1,6,6}
   342: {1,2,2,8}

Crossrefs

The LHS is A061395 (greatest prime index), least A055396.
Without multiplying by 2 in the RHS, we have A106529.
For omega instead of bigomega we have A111907, counted by A239959.
Partitions of this type are counted by A237753.
The RHS is A255201 (twice bigomega).
For mean instead of length we have A361855, counted by A361853.
For median instead of length we have A361856, counted by A361849.
For minimum instead of length we have A361908, counted by A118096.
A001221 (omega) counts distinct prime factors.
A001222 (bigomega) counts prime factors with multiplicity.
A112798 lists prime indices, sum A056239.
A316413 ranks partitions with integer mean, counted by A067538.
A326567/A326568 gives mean of prime indices.
Cf. A067801, A324521, A237752, A237820, A360005, A361205, A361907.

Programs

  • Mathematica
    Select[Range[2,100],PrimePi[FactorInteger[#][[-1,1]]]==2*PrimeOmega[#]&]

A363218 Positive integers whose prime indices satisfy: (length) = 2*(maximum).

Original entry on oeis.org

4, 24, 36, 54, 81, 160, 240, 360, 400, 540, 600, 810, 896, 900, 1000, 1215, 1344, 1350, 1500, 2016, 2025, 2240, 2250, 2500, 3024, 3136, 3360, 3375, 3750, 4536, 4704, 5040, 5600, 5625, 5632, 6250, 6804, 7056, 7560, 7840, 8400, 8448, 9375, 10206, 10584, 10976
Offset: 1

Views

Author

Gus Wiseman, May 23 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.

Examples

			The terms together with their prime indices begin:
      4: {1,1}
     24: {1,1,1,2}
     36: {1,1,2,2}
     54: {1,2,2,2}
     81: {2,2,2,2}
    160: {1,1,1,1,1,3}
    240: {1,1,1,1,2,3}
    360: {1,1,1,2,2,3}
    400: {1,1,1,1,3,3}
    540: {1,1,2,2,2,3}
    600: {1,1,1,2,3,3}
    810: {1,2,2,2,2,3}
    896: {1,1,1,1,1,1,1,4}
    900: {1,1,2,2,3,3}
   1000: {1,1,1,3,3,3}
   1215: {2,2,2,2,2,3}
   1344: {1,1,1,1,1,1,2,4}
   1350: {1,2,2,2,3,3}
   1500: {1,1,2,3,3,3}
   2016: {1,1,1,1,1,2,2,4}
   2025: {2,2,2,2,3,3}
   2240: {1,1,1,1,1,1,3,4}

Crossrefs

The LHS (number of prime indices) is A001222.
The RHS is twice A061395.
Before multiplying by 2 we had A106529.
Partitions of this type are counted by A237753.
For sum instead of length we have A344415, counted by A035363.
An adjoint version is A361909, also counted by A237753.
For minimum instead of maximum we have A363134, counted by A237757.
A112798 lists prime indices, sum A056239.
A326567/A326568 gives mean of prime indices.
Cf. A001221, A067801, A118096, A324521, A237752, A361855, A361908, A363222.

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Length[prix[#]]==2*Max[prix[#]]&]

Formula

Disjoint from A361909.
Showing 1-8 of 8 results.

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