A168656 -id:A168656 - 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

A350890 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 (smallest part) = k*(number of parts).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Seiichi Manyama, Jan 21 2022

Column k is asymptotic to (1 - alfa) * exp(2*sqrt(n*(k*log(alfa)^2 + polylog(2, 1 - alfa)))) * (k*log(alfa)^2 + polylog(2, 1 - alfa))^(1/4) / (2*sqrt(Pi) * sqrt(alfa + 2*k - 2*alfa*k) * n^(3/4)), where alfa is positive real root of the equation alfa^(2*k) + alfa - 1 = 0. - Vaclav Kotesovec, Jan 21 2022

			Triangle begins:
  1;
  0, 1;
  0, 0, 1;
  1, 0, 0, 1;
  1, 0, 0, 0, 1;
  1, 0, 0, 0, 0, 1;
  1, 0, 0, 0, 0, 0, 1;
  1, 1, 0, 0, 0, 0, 0, 1;
  2, 1, 0, 0, 0, 0, 0, 0, 1;
  2, 1, 0, 0, 0, 0, 0, 0, 0, 1;
  3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1;

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

Row sums give A168656.
Column k=1..5 give A006141, A350893, A350894, A350898, A350899.
Cf. A350879, A350889.

T(n, k) = polcoef(sum(i=1, sqrtint(n\k), x^(k*i^2)/prod(j=1, 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 ary[-1] == i * ary.size
    }
  }
  a
end
def A350890(n)
  (1..n).map{|i| A(i)}.flatten
end
p A350890(14)

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

A168657 Number of partitions of n such that the number of parts is divisible by the smallest part.

Original entry on oeis.org

1, 1, 2, 4, 6, 8, 12, 17, 25, 34, 48, 64, 87, 114, 151, 198, 258, 332, 428, 546, 695, 879, 1108, 1388, 1737, 2159, 2680, 3312, 4082, 5009, 6138, 7492, 9126, 11081, 13429, 16228, 19575, 23547, 28277, 33879, 40520, 48354, 57615, 68509, 81337, 96388, 114055

Offset: 1

Vladeta Jovovic, Dec 02 2009

Vaclav Kotesovec, Table of n, a(n) for n = 1..9464 (terms 1..1000 from Alois P. Heinz)

Cf. A079501, A168655, A168656.

b:= proc(n, i, t) option remember;
      `if`(n<1, 0, `if`(i=1, 1, `if`(i<1, 0,
      `if`(irem(n, i)=0 and irem(t+n/i, i)=0, 1, 0)+
            add(b(n-i*j, i-1, t+j), j=0..n/i))))
    end:
a:= n-> b(n, n, 0):
seq(a(n), n=1..60);  # Alois P. Heinz, May 24 2012

b[n_, i_, t_] := b[n, i, t] = If[n<1, 0, If[i==1, 1, If[i<1, 0, If [Mod[n, i]==0 && Mod[t+n/i, i]==0, 1, 0] + Sum[b[n-i*j, i-1, t+j], {j, 0, n/i}]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Jul 01 2015, after Alois P. Heinz *)
Table[Count[IntegerPartitions[n],?(Mod[Length[#],#[[-1]]]==0&)],{n,50}] (* _Harvey P. Dale, Jul 16 2025 *)

my(N=66, x='x+O('x^N)); Vec(sum(i=1, N, sum(j=1, sqrtint(N\i), x^(i*j^2)/prod(k=1, i*j-1, 1-x^k)))) \\ Seiichi Manyama, Jan 21 2022

G.f.: Sum_{n>=1} Sum_{d|n} x^(n*d)/Product_{k=1..n-1}(1-x^k).
G.f.: Sum_{i>=1} Sum_{j>=1} x^(i*j^2)/Product_{k=1..i*j-1} (1-x^k). - Seiichi Manyama, Jan 21 2022
From Vaclav Kotesovec, Oct 17 2024: (Start)
a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*n*sqrt(3)) * (1 - (sqrt(3/2)/Pi + 13*Pi / (2^(7/2) * 3^(3/2))) / sqrt(n)).
A000041(n) - a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(7/2) * n^(3/2)). (End)

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

A350894 Number of partitions of n such that (smallest part) = 3*(number of parts).

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 13, 14, 15, 17, 18, 20, 22, 24, 26, 29, 31, 34, 37, 40, 43, 47, 50, 54, 58, 62, 66, 71, 75, 80, 85, 90, 95, 102, 107, 114, 121, 129, 136, 146, 154, 165, 175, 187, 198, 213

Offset: 1

Seiichi Manyama, Jan 21 2022

nonn

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

Column 3 of A350890.

Cf. A168656, A237756.

my(N=99, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, sqrtint(N\3), x^(3*k^2)/prod(j=1, k-1, 1-x^j))))

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

a(n) ~ (1 - alfa) * exp(2*sqrt(n*(3*log(alfa)^2 + polylog(2, 1 - alfa)))) * (3*log(alfa)^2 + polylog(2, 1 - alfa))^(1/4) / (2*sqrt(Pi) * sqrt(6 - 5*alfa) * n^(3/4)), where alfa = 0.7780895986786010978806823096592944458720784440255... is positive real root of the equation alfa^6 + alfa - 1 = 0. - Vaclav Kotesovec, Jan 22 2022

A350893 Number of partitions of n such that (smallest part) = 2*(number of parts).

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 10, 10, 12, 13, 15, 16, 19, 20, 23, 25, 28, 30, 34, 36, 40, 43, 47, 50, 56, 59, 65, 70, 77, 82, 91, 97, 107, 115, 126, 135, 149, 159, 174, 187, 204, 218, 238, 254, 276, 295, 320, 341, 370, 394, 426, 455, 491, 523, 565

Offset: 1

Seiichi Manyama, Jan 21 2022

nonn

Column 2 of A350890.

Cf. A168656, A237753, A237757.

nmax = 100; Rest[CoefficientList[1 + Series[Sum[x^(2*j^2)*(1 - x^j)/Product[1 - x^i, {i, 1, j}], {j, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jan 21 2022 *)

my(N=99, x='x+O('x^N)); concat(0, Vec(sum(k=1, sqrtint(N\2), x^(2*k^2)/prod(j=1, k-1, 1-x^j))))

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

a(n) ~ (1 - alfa) * exp(2*sqrt(n*(2*log(alfa)^2 + polylog(2, 1 - alfa)))) * (2*log(alfa)^2 + polylog(2, 1 - alfa))^(1/4) / (2*sqrt(Pi) * sqrt(4 - 3*alfa) * n^(3/4)), where alfa = 0.72449195900051561158837228218703656578649448135... is positive real root of the equation alfa^4 + alfa - 1 = 0. - Vaclav Kotesovec, Jan 21 2022

A350898 Number of partitions of n such that (smallest part) = 4*(number of parts).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 17, 17, 19, 20, 22, 23, 26, 27, 30, 32, 35, 37, 41, 43, 47, 50, 54, 57, 62, 65, 70, 74, 79, 83, 89, 93, 99, 104

Offset: 1

Seiichi Manyama, Jan 21 2022

nonn

Column 4 of A350890.

Cf. A168656.

my(N=99, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, sqrtint(N\4), x^(4*k^2)/prod(j=1, k-1, 1-x^j))))

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

a(n) ~ (1 - alfa) * exp(2*sqrt(n*(4*log(alfa)^2 + polylog(2, 1 - alfa)))) * (4*log(alfa)^2 + polylog(2, 1 - alfa))^(1/4) / (2*sqrt(Pi) * sqrt(8 - 7*alfa) * n^(3/4)), where alfa = 0.8116523200278026483934188589034567041719182934245... is positive real root of the equation alfa^8 + alfa - 1 = 0. - Vaclav Kotesovec, Jan 22 2022

A350899 Number of partitions of n such that (smallest part) = 5*(number of parts).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 20, 21, 22, 24, 25, 27, 29, 31, 33, 36, 38, 41, 44, 47

Offset: 1

Seiichi Manyama, Jan 21 2022

nonn

Column 5 of A350890.

Cf. A168656.

my(N=99, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=1, sqrtint(N\5), x^(5*k^2)/prod(j=1, k-1, 1-x^j))))

G.f.: Sum_{k>=1} x^(5*k^2)/Product_{j=1..k-1} (1-x^j).

a(n) ~ (1 - alfa) * exp(2*sqrt(n*(5*log(alfa)^2 + polylog(2, 1 - alfa)))) * (5*log(alfa)^2 + polylog(2, 1 - alfa))^(1/4) / (2*sqrt(Pi) * sqrt(10 - 9*alfa) * n^(3/4)), where alfa = 0.8350790427235590476091499923248865165628469558282... is positive real root of the equation alfa^10 + alfa - 1 = 0. - Vaclav Kotesovec, Jan 22 2022

A171628 Number of compositions of n such that the smallest part is divisible by the number of parts.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 3, 4, 6, 8, 11, 15, 19, 22, 25, 30, 37, 47, 62, 83, 108, 136, 168, 205, 247, 295, 354, 429, 524, 642, 789, 972, 1196, 1466, 1789, 2173, 2625, 3155, 3778, 4515, 5391, 6437, 7692, 9201, 11014, 13186, 15780, 18865, 22516, 26818, 31871, 37791

Offset: 1

Vladeta Jovovic, Dec 13 2009

Alois P. Heinz, Table of n, a(n) for n = 1..900

Cf. A168656, A171625.

b:= proc(n,t,g) option remember; `if` (n=0, `if` (irem(g, t)=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..60); # Alois P. Heinz, Dec 15 2009
A171628 := proc(n) local g,k; g := 0 ; for k from 0 to n do g := g+add (x^(k*d)*(1-x^d)/(1-x)^d,d=numtheory[divisors](k)) ; g := expand(g) ; end do ; coeftayl(g,x=0,n) ; end proc: seq(A171628(n),n=1..60) ; # R. J. Mathar, Dec 14 2009

b[n_, t_, g_] := b[n, t, g] = If[n == 0, If [Mod[g, t] == 0, 1, 0], Sum[b[n - i, t + 1, Min[i, g]], {i, n}]];
a[n_] := b[n, 0, Infinity];
Array[a, 60] (* Jean-François Alcover, May 23 2020, after Alois P. Heinz *)

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

More terms from R. J. Mathar and Alois P. Heinz, Dec 14 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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A350890 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 (smallest part) = k*(number of parts).

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A168657 Number of partitions of n such that the number of parts is divisible by the smallest part.

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

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A350894 Number of partitions of n such that (smallest part) = 3*(number of parts).

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A350893 Number of partitions of n such that (smallest part) = 2*(number of parts).

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Mathematica

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A350898 Number of partitions of n such that (smallest part) = 4*(number of parts).

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A350899 Number of partitions of n such that (smallest part) = 5*(number of parts).

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