A238590 -id:A238590 - cp's OEIS frontend

A238589 Number of partitions p of n such that 2*min(p) is a part of p.

0, 0, 1, 1, 2, 4, 5, 8, 13, 17, 24, 36, 47, 64, 88, 116, 153, 203, 261, 340, 439, 559, 710, 905, 1136, 1427, 1786, 2223, 2756, 3415, 4201, 5167, 6330, 7730, 9413, 11449, 13864, 16767, 20225, 24344, 29228, 35045, 41898, 50029, 59609, 70899, 84165, 99785, 118052

Offset: 1

Clark Kimberling, Mar 01 2014

			a(6) counts these partitions:  42, 321, 2211, 21111.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A117989, A238590, A238591.

Cf. A118096, A238479, A238594.

Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, 2*Min[p]]], {n, 50}]

my(N=50, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, x^(3*k)/prod(j=k, N, 1-x^j)))) \\ Seiichi Manyama, May 17 2023

a(n) = A000041(n) - A238594(n).
G.f.: Sum_{k>=1} x^(3*k)/Product_{j>=k} (1-x^j). - Seiichi Manyama, May 17 2023
a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*n*sqrt(3)) * (1 - (sqrt(3/2)/Pi + 49*Pi/(24*sqrt(6))) / sqrt(n)). - Vaclav Kotesovec, Jun 19 2025

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

Clark Kimberling, Mar 01 2014

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

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)

Cf. A117989, A238589, A238590.

Cf. A237826, A363067.

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

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 *)

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

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)

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

Seiichi Manyama, May 17 2023

nonn

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)

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A117989, A238589, A238590, A238591.

Cf. A237827, A363068.

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

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 *)

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))))

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)

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A238589 Number of partitions p of n such that 2*min(p) is a part of p.

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

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

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