A168657 -id:A168657 - cp's OEIS frontend

A168659 Number of partitions of n such that the number of parts is divisible by the greatest part. Also number of partitions of n such that the greatest part is divisible by the number of parts.

1, 1, 2, 2, 3, 3, 6, 6, 8, 9, 14, 16, 22, 25, 33, 39, 51, 60, 79, 92, 116, 137, 174, 204, 254, 300, 368, 435, 530, 625, 760, 896, 1076, 1267, 1518, 1780, 2121, 2484, 2946, 3444, 4070, 4749, 5594, 6514, 7637, 8879, 10384, 12043, 14040, 16255

Offset: 1

Vladeta Jovovic, Dec 02 2009

nonn

			a(5)=3 because in the partitions [1,1,1,1,1], [1,1,1,2], [1,1,3] the number of parts is divisible by the greatest part; not true for the partitions [1,2,2],[2,3], [1,4], and [5]. - _Emeric Deutsch_, Dec 04 2009
From _Gus Wiseman_, Feb 08 2021: (Start)
The a(1) = 1 through a(10) = 9 partitions of the first type:
  1  11  21   22    311    321     322      332       333        4222
         111  1111  2111   2211    331      2222      4221       4321
                    11111  111111  2221     4211      4311       4411
                                   4111     221111    51111      52111
                                   211111   311111    222111     222211
                                   1111111  11111111  321111     322111
                                                      21111111   331111
                                                      111111111  22111111
                                                                 1111111111
The a(1) = 1 through a(11) = 14 partitions of the second type (A=10, B=11):
  1   2   3    4    5     6     7      8      9       A       B
          21   22   41    42    43     44     63      64      65
                    311   321   61     62     81      82      83
                                322    332    333     622     A1
                                331    611    621     631     632
                                4111   4211   4221    4222    641
                                              4311    4321    911
                                              51111   4411    4322
                                                      52111   4331
                                                              4421
                                                              8111
                                                              52211
                                                              53111
                                                              611111
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 1..5000 (terms 1..301 from Vladeta Jovovic corrected by N. J. A. Sloane, Oct 05 2010, terms 302..1000 from Seiichi Manyama)

Cf. A168656, A168657, A079501, A168655.
Note: A-numbers of Heinz-number sequences are in parentheses below.
The case of equality is A047993 (A106529).
The Heinz numbers of these partitions are A340609/A340610.
If all parts (not just the greatest) are divisors we get A340693 (A340606).
The strict case in the second interpretation is A340828 (A340856).
A006141 = partitions whose length equals their minimum (A324522).
A067538 = partitions whose length/max divides their sum (A316413/A326836).
A200750 = partitions with length coprime to maximum (A340608).
Cf. A003114, A039900, A064173, A064174, A143773, A326837, A340653, A340830, A340851, A340853.
Row sums of A350879.

a := proc (n) local pn, ct, j: with(combinat): pn := partition(n): ct := 0: for j to numbpart(n) do if `mod`(nops(pn[j]), max(seq(pn[j][i], i = 1 .. nops(pn[j])))) = 0 then ct := ct+1 else end if end do: ct end proc: seq(a(n), n = 1 .. 50); # Emeric Deutsch, Dec 04 2009

Table[Length[Select[IntegerPartitions[n],Divisible[Length[#],Max[#]]&]],{n,30}] (* Gus Wiseman, Feb 08 2021 *)
nmax = 100; s = 0; Do[s += Normal[Series[Sum[x^((m+1)*k - 1) * Product[(1 - x^(m*k + j - 1))/(1 - x^j), {j, 1, k-1}], {k, 1, (1 + nmax)/(1 + m) + 1}], {x, 0, nmax}]], {m, 1, nmax}]; Rest[CoefficientList[s, x]] (* Vaclav Kotesovec, Oct 18 2024 *)

G.f.: Sum_{i>=1} Sum_{j>=1} x^((i+1)*j-1) * Product_{k=1..j-1} (1-x^(i*j+k-1))/(1-x^k). - Seiichi Manyama, Jan 24 2022

a(n) ~ c * exp(Pi*sqrt(2*n/3)) / n^(3/2), where c = 0.04628003... - Vaclav Kotesovec, Nov 16 2024

Extended by Emeric Deutsch, Dec 04 2009

A168656 Number of partitions of n such that the smallest part is divisible by the number of parts.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 5, 6, 7, 7, 8, 10, 11, 13, 15, 18, 20, 23, 25, 29, 33, 36, 41, 47, 53, 58, 66, 74, 83, 92, 103, 116, 130, 144, 160, 179, 199, 219, 243, 269, 298, 328, 362, 399, 441, 484, 533, 586, 645, 708, 778, 854, 937, 1026, 1124, 1230, 1347, 1470, 1607, 1756, 1917, 2089

Offset: 1

Vladeta Jovovic, Dec 01 2009, Dec 04 2009

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000041, A006141, A073336, A079501, A168655, A168657, A168659.

nmax = 100; Rest[CoefficientList[Series[Sum[x^(k^2)/((1 - x^(k^2))*Product[1 - x^j, {j, 1, k-1}]), {k, 1, Sqrt[nmax]}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Oct 16 2024 *)
Table[Count[IntegerPartitions[n],?(Mod[#[[-1]],Length[#]]==0&)],{n,70}] (* _Harvey P. Dale, Dec 22 2024 *)

N=100; x='x+O('x^N);
Vec( sum(k=1,sqrtint(N), x^(k^2)/(1-x^(k^2)) / prod(i=1,k-1, 1-x^i) ) )

G.f.: Sum_{k>=1} x^(k^2)/((1-x^(k^2)) * Product_{i=1..k-1} (1-x^i)).

a(n) ~ c * exp(2*Pi*sqrt(n/15)) / n^(3/4), where c = 1 / (2 * 3^(1/4) * sqrt(5) * phi^(3/2)) = 0.08255116908... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Oct 17 2024

A350889 Triangle T(n,k), n >= 1, 1 <= k <= n, read by rows, where T(n,k) is the number of partitions of n such that k*(smallest part) = (number of parts).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 3, 2, 1, 1, 1, 2, 3, 4, 3, 2, 1, 1, 2, 2, 4, 5, 5, 3, 2, 1, 1, 2, 3, 4, 7, 6, 5, 3, 2, 1, 1, 3, 4, 5, 8, 9, 7, 5, 3, 2, 1, 1, 3, 5, 6, 10, 11, 10, 7, 5, 3, 2, 1, 1, 4, 6, 7, 12, 15, 13, 11, 7, 5, 3, 2, 1, 1, 4, 8, 8, 14, 18, 18, 14, 11, 7, 5, 3, 2, 1, 1

Offset: 1

Seiichi Manyama, Jan 21 2022

Column k is asymptotic to r^2 * (k*log(r)^2 + polylog(2, r^2))^(1/4) * exp(2*sqrt((k*log(r)^2 + polylog(2, r^2))*n)) / (2*sqrt(Pi*k*(k - (k-2)*r^2)) * n^(3/4)), where r is the positive real root of the equation r^2 = 1 - r^k. - Vaclav Kotesovec, Oct 14 2024

			Triangle begins:
  1;
  0, 1;
  0, 1, 1;
  1, 1, 1, 1;
  1, 1, 2, 1, 1;
  1, 1, 2, 2, 1, 1;
  1, 1, 3, 3, 2, 1, 1;
  1, 2, 3, 4, 3, 2, 1, 1;
  2, 2, 4, 5, 5, 3, 2, 1, 1;
  2, 3, 4, 7, 6, 5, 3, 2, 1, 1;
  3, 4, 5, 8, 9, 7, 5, 3, 2, 1, 1;

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50).

Row sums give A168657.
Column k=1..8 give A006141, A237757, A350892, A350896, A350897, A377076, A377077, A377075.
Cf. A350879, A350890.

T(n, k) = polcoef(sum(i=1, sqrtint(n\k), x^(k*i^2)/prod(j=1, k*i-1, 1-x^j+x*O(x^n))), n);

def partition(n, min, max)
  return [[]] if n == 0
  [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
end
def A(n)
  a = Array.new(n, 0)
  partition(n, 1, n).each{|ary|
    (1..n).each{|i|
      a[i - 1] += 1 if i * ary[-1] == ary.size
    }
  }
  a
end
def A350889(n)
  (1..n).map{|i| A(i)}.flatten
end
p A350889(14)

G.f. of column k: Sum_{i>=1} x^(k*i^2)/Product_{j=1..k*i-1} (1-x^j).

A079501 Number of compositions of the integer n with strictly smallest part in the first position.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 8, 12, 19, 28, 45, 70, 110, 173, 275, 436, 695, 1107, 1769, 2831, 4537, 7276, 11683, 18774, 30194, 48592, 78247, 126062, 203192, 327645, 528518, 852815, 1376491, 2222294, 3588628, 5796196, 9363458, 15128631, 24447014, 39510108

Offset: 1

Arnold Knopfmacher, Jan 21 2003

nonn

Also number of compositions of n such that the first part is divisible by the number of parts . [Vladeta Jovovic, Dec 02 2009]

			The a(9)=19 such compositions of 9 are
[ 1]  [ 1 2 2 2 2 ]
[ 2]  [ 1 2 2 4 ]
[ 3]  [ 1 2 3 3 ]
[ 4]  [ 1 2 4 2 ]
[ 5]  [ 1 2 6 ]
[ 6]  [ 1 3 2 3 ]
[ 7]  [ 1 3 3 2 ]
[ 8]  [ 1 3 5 ]
[ 9]  [ 1 4 2 2 ]
[10]  [ 1 4 4 ]
[11]  [ 1 5 3 ]
[12]  [ 1 6 2 ]
[13]  [ 1 8 ]
[14]  [ 2 3 4 ]
[15]  [ 2 4 3 ]
[16]  [ 2 7 ]
[17]  [ 3 6 ]
[18]  [ 4 5 ]
[19]  [ 9 ]
- _Joerg Arndt_, Jan 01 2013

Arnold Knopfmacher and Neville Robbins, Compositions with parts constrained by the leading summand, Ars Combin. 76 (2005), 287-295.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A168655, A168656, A168657. [From Vladeta Jovovic, Dec 02 2009]

b:= proc(n, s) option remember; `if`(n=0, 1, add(
      `if`(n-j>0 and n-j<=s, 0, b(n-j, s)), j=s+1..n))
    end:
a:= n-> 1 +add(b(n-j, j), j=1..n/2):
seq(a(n), n=1..60);  # Alois P. Heinz, Apr 29 2014

b[n_, s_] := b[n, s] = If[n == 0, 1, Sum[ If[n - j > 0 && n - j <= s, 0, b[n - j, s]], {j, s + 1, n}]]; a[n_] := 1 + Sum[b[n - j, j], {j, 1, n/2}]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Jan 16 2015, after Alois P. Heinz *)

G.f.: Sum_{k>=1} (1-z)*z^k/(1-z-z^(k+1)).
G.f.: Sum_{k>=1} z^(2*k-1)/((1-z^k)*(1-z)^(k-1)), cf. A105039. - Vladeta Jovovic, Apr 05 2005
a(n) ~ 1/sqrt(5) * ((1+sqrt(5))/2)^(n-2). - Vaclav Kotesovec, May 01 2014
G.f.: Sum_{n>=1} q^n/(1-Sum_{k>=n+1} q^k). - Joerg Arndt, Jan 03 2024

More terms from Benoit Cloitre, Jan 21 2003

A350892 Number of partitions of n such that 3*(smallest part) = (number of parts).

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 12, 15, 18, 22, 27, 33, 40, 48, 58, 69, 82, 98, 115, 135, 158, 184, 214, 248, 286, 330, 379, 435, 497, 569, 648, 739, 840, 955, 1082, 1228, 1388, 1572, 1775, 2005, 2259, 2549, 2867, 3228, 3626, 4076, 4571, 5131, 5745, 6438, 7199, 8053, 8992, 10045, 11199

Offset: 1

Seiichi Manyama, Jan 21 2022

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..20000

Column 3 of A350889.

Cf. A168657, A237756, A350894.

CoefficientList[Series[Sum[x^(3k^2)/Product[1-x^j,{j,3k-1}],{k,64}],{x,0,64}],x] (* Stefano Spezia, Jan 22 2022 *)
Table[Count[IntegerPartitions[n],?(3#[[-1]]==Length[#]&)],{n,70}] (* _Harvey P. Dale, Jul 13 2023 *)

my(N=66, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, sqrtint(N\3), x^(3*k^2)/prod(j=1, 3*k-1, 1-x^j))))

G.f.: Sum_{k>=1} x^(3*k^2)/Product_{j=1..3*k-1} (1-x^j).

a(n) ~ c * exp(2*sqrt((5*log(A075778)^2 + 2*polylog(2, 1 - A075778))*n)) / n^(3/4), where c = (3*log(A075778)^2 + polylog(2, A075778^2))^(1/4) / (2*sqrt(3*Pi*(1 + A075778)*(2 + 3*A075778))) = 0.0582980106266835787... - Vaclav Kotesovec, Jan 24 2022, updated Oct 14 2024

A350896 Number of partitions of n such that 4*(smallest part) = (number of parts).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 20, 22, 26, 30, 35, 40, 48, 55, 65, 76, 90, 105, 126, 147, 175, 206, 244, 286, 339, 396, 467, 545, 638, 741, 865, 1000, 1160, 1337, 1543, 1770, 2035, 2325, 2660, 3029, 3451, 3916, 4447, 5029, 5691, 6419, 7242, 8146, 9167, 10286, 11546, 12930, 14481, 16185

Offset: 1

Seiichi Manyama, Jan 21 2022

nonn

			For n=7 there are a(7)=3 such partitions: [1,2,2,2], [1,1,2,3] and [1,1,1,4]. - _R. J. Mathar_, Jun 20 2022

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

Column 4 of A350889.

Cf. A168657.

CoefficientList[Series[Sum[x^(4k^2)/Product[1-x^j,{j,4k-1}],{k,63}],{x,0,63}],x] (* Stefano Spezia, Jan 22 2022 *)

my(N=66, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, sqrtint(N\4), x^(4*k^2)/prod(j=1, 4*k-1, 1-x^j))))

G.f.: Sum_{k>=1} x^(4*k^2)/Product_{j=1..4*k-1} (1-x^j).

a(n) ~ c * exp(Pi*sqrt(2*n/5)) / n^(3/4), where c = (3 - sqrt(5))^(1/4) / (8*sqrt(5)) = 0.05226232058... - Vaclav Kotesovec, Jan 25 2022, updated Oct 13 2024

A350897 Number of partitions of n such that 5*(smallest part) = (number of parts).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 5, 6, 9, 11, 15, 18, 23, 27, 34, 39, 47, 55, 65, 74, 87, 99, 115, 131, 151, 172, 199, 226, 260, 298, 343, 393, 454, 522, 603, 696, 804, 929, 1076, 1243, 1438, 1664, 1924, 2222, 2567, 2961, 3413, 3931, 4520, 5193, 5959, 6827, 7811, 8928, 10186, 11607, 13208, 15008, 17028, 19297

Offset: 1

Seiichi Manyama, Jan 21 2022

nonn

In general, for m >= 1, if g.f.= Sum_{k>=1} x^(m*k^2)/Product_{j=1..m*k-1} (1-x^j), then a(n) ~ r^2 * (m*log(r)^2 + polylog(2, r^2))^(1/4) * exp(2*sqrt((m*log(r)^2 + polylog(2, r^2))*n)) / (2*sqrt(Pi*m*(m - (m-2)*r^2)) * n^(3/4)), where r is the positive real root of the equation r^2 = 1 - r^m. - Vaclav Kotesovec, Oct 14 2024

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

Column 5 of A350889.

Cf. A168657.

CoefficientList[Series[Sum[x^(5k^2)/Product[1-x^j,{j,5k-1}],{k,62}],{x,0,62}],x] (* Stefano Spezia, Jan 22 2022 *)

my(N=66, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=1, sqrtint(N\5), x^(5*k^2)/prod(j=1, 5*k-1, 1-x^j))))

G.f.: Sum_{k>=1} x^(5*k^2)/Product_{j=1..5*k-1} (1-x^j).
a(n) ~ c * exp(Pi*sqrt(r*n)) / n^(3/4), where r = 0.42067169741517... and c = 0.04778365700734... - Vaclav Kotesovec, Jan 26 2022
a(n) ~ r^2 * (5*log(r)^2 + polylog(2, r^2))^(1/4) * exp(2*sqrt((5*log(r)^2 + polylog(2, r^2))*n)) / (2*sqrt(5*Pi*(5 - 3*r^2)) * n^(3/4)), where r = 0.808730600479392... is the real root of the equation r^2 = 1 - r^5. - Vaclav Kotesovec, Oct 14 2024

A171625 Number of compositions of n such that the number of parts is divisible by the smallest part.

Original entry on oeis.org

1, 1, 3, 7, 15, 29, 58, 118, 242, 493, 997, 2005, 4024, 8071, 16183, 32439, 65003, 130214, 260768, 522084, 1045045, 2091489, 4185209, 8373979, 16753651, 33516419, 67047467, 134118462, 268274858, 536611011, 1073321222, 2146803124, 4293866550, 8588154649

Offset: 1

Vladeta Jovovic, Dec 13 2009

Alois P. Heinz, Table of n, a(n) for n = 1..500

Cf. A168657.

b:= proc(n,t,g) option remember; `if` (n=0, `if` (irem(t, g)=0, 1, 0), add (b(n-i, t+1, min(i, g)), i=1..n)) end: a:= n-> b(n, 0, infinity): seq (a(n), n=1..40);  # Alois P. Heinz, Dec 15 2009

a[n_] := SeriesCoefficient[ Sum[(1-x^k)/(1-x)^k*Sum[x^(k*d), {d, Divisors[k]}], {k, 0, n}], {x, 0, n}]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Feb 24 2015 *)

G.f.: Sum_{n>=0}[(1-x^n)/(1-x)^n*Sum_{d|n}x^(n*d)].

a(n) ~ 2^(n-1). - Vaclav Kotesovec, May 01 2014

More terms from Alois P. Heinz, Dec 15 2009

cp's OEIS Frontend

A168659 Number of partitions of n such that the number of parts is divisible by the greatest part. Also number of partitions of n such that the greatest part is divisible by the number of parts.

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A168656 Number of partitions of n such that the smallest part is divisible by the number of parts.

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A350889 Triangle T(n,k), n >= 1, 1 <= k <= n, read by rows, where T(n,k) is the number of partitions of n such that k*(smallest part) = (number of parts).

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A079501 Number of compositions of the integer n with strictly smallest part in the first position.

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A350892 Number of partitions of n such that 3*(smallest part) = (number of parts).

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A350896 Number of partitions of n such that 4*(smallest part) = (number of parts).

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A350897 Number of partitions of n such that 5*(smallest part) = (number of parts).

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