A117989 -id:A117989 - cp's OEIS frontend

A363485 Number of integer partitions of n covering an initial interval of positive integers with more than one mode.

0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 2, 1, 3, 1, 2, 6, 5, 3, 8, 4, 8, 11, 13, 9, 17, 17, 19, 25, 24, 23, 44, 35, 39, 54, 55, 63, 83, 79, 86, 104, 119, 125, 157, 164, 178, 220, 237, 251, 297, 324, 357, 413, 439, 486, 562, 607, 673, 765, 828, 901, 1040, 1117, 1220

Offset: 0

Gus Wiseman, Jun 06 2023

nonn

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.

			The a(n) partitions for n = {3, 6, 12, 15, 16, 18}:
  (21)  (321)   (332211)    (54321)       (443221)    (4433211)
        (2211)  (3222111)   (433221)      (3332221)   (5432211)
                (22221111)  (443211)      (4332211)   (43332111)
                            (33222111)    (33322111)  (333222111)
                            (322221111)   (43222111)  (333321111)
                            (2222211111)              (3322221111)
                                                      (32222211111)
                                                      (222222111111)

For parts instead of multiplicities we have A025147, complement A096765.
For co-mode we have A363264, complement A363263.
The complement is counted by A363484.
A000041 counts integer partitions, A000009 covering an initial interval.
A071178 counts maxima in prime factorization, modes A362611.
A362607 counts partitions with multiple modes, co-modes A362609.
A362608 counts partitions with a unique mode, co-mode A362610.
A362614 counts partitions by number of modes, co-modes A362615.
Cf. A002865, A008284, A105039, A117989, A243737, A362612.

Table[If[n==0,0,Length[Select[IntegerPartitions[n], Union[#]==Range[Max@@#]&&Length[Commonest[#]]>1&]]],{n,0,30}]

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)

A173870 Consider each term k contained in A114994; write 2k if and only if 2k is not a member of A114994.

Original entry on oeis.org

6, 14, 20, 22, 30, 38, 46, 62, 70, 72, 78, 84, 86, 94, 126, 134, 142, 148, 150, 158, 174, 190, 254, 262, 270, 272, 276, 278, 286, 294, 302, 318, 340, 342, 350, 382, 510, 518, 526, 532, 534, 542, 550, 558, 574, 584, 590, 596, 598, 606, 638, 686, 702, 766, 1022

Offset: 0

Alford Arnold, Mar 01 2010

Recall that A114994 can be regarded as a table with row lengths A000041(n).
Likewise, a(n) has row lengths 0,1,1,3,3,7,8,14,18,28,35,53,67,... which appears to coincide with sequence A117989.
The row lengths also match 1 2 3 5 7 11 15 22 ... minus 1 1 2 2 4 4 7 8 ... - Alford Arnold, Mar 30 2010

			Row three of A114994 is 4,5,7 when doubled becomes 8,10,14.
8 and 10 are in A114994 so not in a(n); 14 is not in A114994 so is in a(n).

Cf. A000041 A114994 A117989 (A125106, A126441, A161924)(3 closely related sequences).

Cf. A002865 [From Alford Arnold, Mar 30 2010]

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A363485 Number of integer partitions of n covering an initial interval of positive integers with more than one mode.

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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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A173870 Consider each term k contained in A114994; write 2k if and only if 2k is not a member of A114994.

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