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 1-5 of 5 results.

A325534 Number of separable partitions of n; see Comments.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 6, 10, 14, 19, 26, 37, 49, 66, 87, 116, 152, 198, 254, 329, 422, 536, 678, 858, 1077, 1349, 1681, 2089, 2587, 3193, 3927, 4820, 5897, 7191, 8749, 10623, 12861, 15535, 18724, 22518, 27029, 32373, 38697, 46174, 54998, 65382, 77601, 91950, 108777
Offset: 0

Views

Author

Clark Kimberling, May 08 2019

Keywords

Comments

Definition: a partition is separable if there is an ordering of its parts in which no consecutive parts are identical; otherwise the partition is inseparable.
A partition with k parts is separable if and only if there is no part whose multiplicity is greater than ceiling(k/2). - Andrew Howroyd, Jan 31 2024

Examples

			For n=5, the partition 1+2+2 is separable as 2+1+2, and 2+1+1+1 is inseparable.

Links

Crossrefs

Cf. A000041, A238589, A325535.

Programs

  • Mathematica
    Table[Length[Select[Map[Quotient[(1 + Length[#]), Max[Map[Length, Split[#]]]] &,
IntegerPartitions[nn]], # > 1 &]], {nn, 50}]  (* Peter J. C. Moses, May 07 2019 *)

Formula

a(n) = A000041(n) - A325535(n).

Extensions

a(0)=1 prepended by Alois P. Heinz, Jan 20 2024

A238590 Number of partitions p of n such that 3*min(p) is a part of p.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 6, 7, 12, 16, 25, 32, 46, 61, 86, 110, 149, 192, 257, 326, 425, 538, 694, 871, 1107, 1381, 1740, 2154, 2689, 3313, 4103, 5024, 6176, 7529, 9201, 11157, 13554, 16365, 19784, 23782, 28610, 34260, 41039, 48958, 58405, 69431, 82525, 97775
Offset: 1

Views

Author

Clark Kimberling, Mar 01 2014

Keywords

Examples

			a(7) = 3 counts these partitions:  331, 3211, 31111.

Links

Crossrefs

Cf. A117989, A238589, A238591.
Cf. A237825, A363066.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i>n, 0, b(n, i+1)+b(n-i, i)))
    end:
a:= n-> add(b(n-4*i, i), i=1..n/4):
seq(a(n), n=1..60);  # Alois P. Heinz, Mar 03 2014
  • Mathematica
    Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, 3*Min[p]]], {n, 50}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i>n, 0, b[n, i+1] + b[n-i, i]]];
a[n_] := Sum[b[n-4i, i], {i, 1, n/4}];
Array[a, 60] (* Jean-François Alcover, Jun 04 2021, after Alois P. Heinz *)
  • PARI
    my(N=50, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(4*k)/prod(j=k, N, 1-x^j)))) \\ Seiichi Manyama, May 17 2023

Formula

G.f.: Sum_{k>=1} x^(4*k)/Product_{j>=k} (1-x^j). - Seiichi Manyama, May 17 2023
From Vaclav Kotesovec, Jun 19 2025: (Start)
a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n) * (1 - (sqrt(3/2)/Pi + 73*Pi/(24*sqrt(6))) / sqrt(n)).
A000041(n) - a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (2^(5/2) * n^(3/2)). (End)

A238591 Number of partitions p of n such that 4*min(p) is a part of p.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 11, 16, 23, 32, 45, 60, 81, 109, 144, 190, 247, 320, 412, 529, 675, 854, 1078, 1355, 1695, 2117, 2626, 3251, 4010, 4932, 6047, 7394, 9012, 10959, 13290, 16083, 19407, 23379, 28090, 33689, 40317, 48158, 57406, 68324, 81155, 96248
Offset: 1

Views

Author

Clark Kimberling, Mar 01 2014

Keywords

Examples

			a(9) = 5 counts these partitions:  441, 4311, 4221, 42111, 411111.

Links

Crossrefs

Cf. A117989, A238589, A238590.
Cf. A237826, A363067.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i>n, 0, b(n, i+1)+b(n-i, i)))
    end:
a:= n-> add(b(n-5*i, i), i=1..n/5):
seq(a(n), n=1..60);  # Alois P. Heinz, Mar 03 2014
  • Mathematica
    Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, 4*Min[p]]], {n, 50}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i>n, 0, b[n, i+1] + b[n-i, i]]];
a[n_] := Sum[b[n - 5*i, i], {i, 1, n/5}];
Array[a, 60] (* Jean-François Alcover, Jun 09 2021, after Alois P. Heinz *)
  • PARI
    my(N=50, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=1, N, x^(5*k)/prod(j=k, N, 1-x^j)))) \\ Seiichi Manyama, May 17 2023

Formula

G.f.: Sum_{k>=1} x^(5*k)/Product_{j>=k} (1-x^j). - Seiichi Manyama, May 17 2023
From Vaclav Kotesovec, Jun 19 2025: (Start)
a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n) * (1 - (sqrt(3/2)/Pi + 97*Pi/(24*sqrt(6))) / sqrt(n)).
A000041(n) - a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3*sqrt(2)*n^(3/2)). (End)

A238594 Number of partitions p of n such that 2*min(p) is not a part of p.

Original entry on oeis.org

1, 2, 2, 4, 5, 7, 10, 14, 17, 25, 32, 41, 54, 71, 88, 115, 144, 182, 229, 287, 353, 443, 545, 670, 822, 1009, 1224, 1495, 1809, 2189, 2641, 3182, 3813, 4580, 5470, 6528, 7773, 9248, 10960, 12994, 15355, 18129, 21363, 25146, 29525, 34659, 40589, 47488, 55473
Offset: 1

Views

Author

Clark Kimberling, Mar 01 2014

Keywords

Comments

a(n) is also the number of partitions of n with a part whose multiplicity is greater than half the total number of parts. - Andrew Howroyd, Jan 17 2024

Examples

			a(6) counts all 11 partitions of 6 except these: 42, 321, 2211, 21111.

Links

Crossrefs

Cf. A238589, A325535.

Programs

  • Mathematica
    Table[Count[IntegerPartitions[n], p_ /; !MemberQ[p, 2*Min[p]]], {n, 50}]
  • PARI
    seq(n) = {Vec(sum(k=1, n\2+1, x^(2*k-2)*(1 + x - x^(k-1))/prod(j=1, k, 1 - x^j, 1 + O(x^(n-2*k+3))), O(x*x^n)))} \\ Andrew Howroyd, Jan 17 2024

Formula

a(n) = A000041(n) - A238589(n).
a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3*2^(3/2)*n^(3/2)). - Vaclav Kotesovec, Jun 09 2021
a(n) = Sum_{k>=1} x^(2*k-2)*(1 + x - x^(k-1))/(Product_{j=1..k} (1 - x^j)). - Andrew Howroyd, Jan 17 2024

A361459 Number of partitions p of n such that 5*min(p) is a part of p.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 12, 15, 23, 31, 44, 58, 82, 105, 142, 185, 244, 312, 409, 516, 664, 837, 1063, 1328, 1674, 2074, 2588, 3194, 3952, 4847, 5964, 7270, 8884, 10786, 13104, 15832, 19147, 23027, 27709, 33203, 39776, 47476, 56661, 67382, 80108, 94960, 112494, 132919, 156965
Offset: 1

Views

Author

Seiichi Manyama, May 17 2023

Keywords

Comments

From Vaclav Kotesovec, Jun 19 2025: (Start)
In general, for m>1, if g.f. = Sum_{k>=0} x^(m*k) / Product_{j>=k} (1 - x^j), then A000041(n) - a(n) ~ Pi * (m-1) * exp(Pi*sqrt(2*n/3)) / (3*2^(5/2)*n^(3/2)).
a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*n*sqrt(3)) * (1 - (sqrt(3/2)/Pi + (m - 23/24)*Pi / sqrt(6)) / sqrt(n)). (End)

Links

Crossrefs

Cf. A117989, A238589, A238590, A238591.
Cf. A237827, A363068.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i>n, 0, b(n, i+1)+b(n-i, i)))
    end:
a:= n-> add(b(n-6*i, i), i=1..n/6):
seq(a(n), n=1..60);  # Alois P. Heinz, May 17 2023
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, 1, If[i > n, 0, b[n, i+1] + b[n-i, i]]];
a[n_] := Sum[b[n - 6 i, i], {i, 1, n/6}];
Array[a, 60] (* Jean-François Alcover, May 30 2024, after Alois P. Heinz *)
  • PARI
    my(N=60, x='x+O('x^N)); concat([0, 0, 0, 0, 0], Vec(sum(k=1, N, x^(6*k)/prod(j=k, N, 1-x^j))))

Formula

G.f.: Sum_{k>=1} x^(6*k)/Product_{j>=k} (1-x^j).
From Vaclav Kotesovec, Jun 19 2025: (Start)
a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n) * (1 - (sqrt(3/2)/Pi + 121*Pi/(24*sqrt(6))) / sqrt(n)).
A000041(n) - a(n) ~ 5 * Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(5/2) * n^(3/2)). (End)
Showing 1-5 of 5 results.

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