A079501 -id:A079501 - 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

A105039 Number of compositions of n with unique smallest part.

Original entry on oeis.org

1, 1, 3, 3, 8, 11, 20, 34, 59, 96, 167, 282, 475, 800, 1352, 2275, 3828, 6426, 10785, 18085, 30297, 50698, 84770, 141623, 236425, 394381, 657380, 1094975, 1822628, 3031843, 5040129, 8373594, 13903588, 23072567, 38267330, 63435438, 105103059, 174054820, 288105394

Offset: 1

Vladeta Jovovic, Apr 03 2005

			a(5) = 8 because we have 5, 14, 41, 23, 32, 122, 212 and 221.

Alois P. Heinz, Table of n, a(n) for n = 1..2000 (first 1000 terms from Vincenzo Librandi)

Cf. A079501, A097979.

Column k=1 of A238342.

G:= sum(k*x^(2*k-1)/((1-x^k)*(1-x)^(k-1)), k=1..70): Gser:=series(G,x=0,44): seq(coeff(Gser,x^n),n=1..41); # Emeric Deutsch, Apr 13 2005

nn=37;Drop[CoefficientList[Series[Sum[x^j/(1-x^(j+1)/(1-x))^2,{j,1,nn}],{x,0,nn}],x],1] (* Geoffrey Critzer, Mar 31 2014 *)

a(n)=1+sum(k=2,(n+3)\2,k*sum(s=1,(n-1)\k,binomial(n-k*s-1,k-2))) (Alekseyev)

G.f.: Sum_{k>0} k * x^(2*k-1)/((1 - x^k) * (1 - x)^(k-1)) = (1 - x)^2 * Sum_{k>0} x^k/(1 - x - x^(k+1))^2.
a(n) = 1 + sum(k=2..[(n+3)/2], k * sum(s=1..[(n-1)/k], binomial(n-k*s-1, k-2) ) ). - Max Alekseyev, Apr 15 2005
a(n) ~ (2*sqrt(5)-4)/10 * n * ((1+sqrt(5))/2)^n. - Vaclav Kotesovec, May 02 2014
Equivalently, a(n) ~ n * phi^(n-3) / 5, where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 07 2021

More terms from Emeric Deutsch and Max Alekseyev, Apr 13 2005

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

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)

A168655 Number of compositions such that the number of parts is divisible by the first part.

Original entry on oeis.org

1, 1, 3, 5, 11, 22, 44, 88, 177, 355, 710, 1419, 2838, 5679, 11363, 22727, 45443, 90862, 181703, 363419, 726903, 1453875, 2907667, 5814880, 11628864, 23256828, 46513965, 93031069, 186068503, 372142797, 744280096, 1488527555, 2976987042, 5953897971, 11907811651

Offset: 1

Vladeta Jovovic, Dec 01 2009

Alois P. Heinz, Table of n, a(n) for n = 1..250

Cf. A079501.

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

A101510[n_] := Sum[If[Mod[i+1, k+1] == 0, Binomial[n-k, i], 0], {k, 0, n/2}, {i, 0, n-k}]; A168655 =  Join[{1}, Table[A101510[n], {n, 0, 32}] // Differences] (* Jean-François Alcover, Jan 24 2014 *)

G.f.: (1-x)*Sum(x^(2*n-1)/((1-x)^n-x^n),n=1..infinity), First differences of A101510.

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

More terms from Alois P. Heinz, Dec 15 2009

A224960 Number of compositions [p(1), p(2), ..., p(k)] of n such that p(j) >= p(1) - 1.

Original entry on oeis.org

1, 1, 2, 4, 7, 14, 26, 52, 101, 200, 396, 787, 1564, 3117, 6214, 12398, 24749, 49427, 98740, 197303, 394323, 788201, 1575695, 3150265, 6298732, 12594595, 25184598, 50361842, 100711888, 201404839, 402779246, 805509560, 1610940381, 3221753990

Offset: 0

Joerg Arndt, Apr 21 2013

nonn

			The a(5) = 14 such compositions of 5 are
01:  [ 1 1 1 1 1 ]
02:  [ 1 1 1 2 ]
03:  [ 1 1 2 1 ]
04:  [ 1 1 3 ]
05:  [ 1 2 1 1 ]
06:  [ 1 2 2 ]
07:  [ 1 3 1 ]
08:  [ 1 4 ]
09:  [ 2 1 1 1 ]
10:  [ 2 1 2 ]
11:  [ 2 2 1 ]
12:  [ 2 3 ]
13:  [ 3 2 ]
14:  [ 5 ]
(the two forbidden compositions are [ 3 1 1 ] and [ 4 1 ]).

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A171682 (compositions such that p(j) >= p(1)).
Cf. A079501 (compositions such that p(j) > p(1)).
Cf. A048888 (compositions such that p(j) <= p(1) + 1).
Cf. A007059 (compositions such that p(j) < p(1)).
Cf. A079500 (compositions such that p(j) <= p(1)).

b:= proc(n, i) option remember; `if`(n=0, 1, add(b(n-j,
      `if`(i=0, max(1, j-1), i)), j=`if`(i=0, 1, i)..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, May 02 2013

b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n - j, If[i == 0, Max[1, j - 1], i]], {j, If[i == 0, 1, i], n}]];
a[n_] := b[n, 0];
a /@ Range[0, 50] (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz *)

a(n) ~ 3 * 2^(n-3). - Vaclav Kotesovec, May 01 2014

A238348 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n where the k-th part is the last occurrence of a smallest part, n>=1, 1<=k<=n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 3, 2, 1, 4, 4, 4, 3, 1, 5, 7, 8, 7, 4, 1, 8, 11, 14, 14, 11, 5, 1, 12, 18, 23, 27, 25, 16, 6, 1, 19, 27, 39, 49, 51, 41, 22, 7, 1, 28, 44, 64, 85, 98, 92, 63, 29, 8, 1, 45, 69, 103, 144, 180, 189, 155, 92, 37, 9, 1, 70, 109, 166, 241, 319, 366, 344, 247, 129, 46, 10, 1

Offset: 1

Joerg Arndt and Alois P. Heinz, Feb 25 2014

Column k=1 is A079501.

Row sums are A011782.

			Triangle starts:
01:  1,
02:  1, 1,
03:  2, 1, 1,
04:  2, 3, 2, 1,
05:  4, 4, 4, 3, 1,
06:  5, 7, 8, 7, 4, 1,
07:  8, 11, 14, 14, 11, 5, 1,
08:  12, 18, 23, 27, 25, 16, 6, 1,
09:  19, 27, 39, 49, 51, 41, 22, 7, 1,
10:  28, 44, 64, 85, 98, 92, 63, 29, 8, 1,
11:  45, 69, 103, 144, 180, 189, 155, 92, 37, 9, 1,
12:  70, 109, 166, 241, 319, 366, 344, 247, 129, 46, 10, 1,
13:  110, 172, 267, 398, 551, 679, 709, 591, 376, 175, 56, 11, 1,
...

Joerg Arndt, Table of n, a(n) for n = 1..465 (rows 1..30, flattened)

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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A105039 Number of compositions of n with unique smallest part.

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

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A168655 Number of compositions such that the number of parts is divisible by the first part.

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Maple

Mathematica

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A224960 Number of compositions [p(1), p(2), ..., p(k)] of n such that p(j) >= p(1) - 1.

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A238348 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n where the k-th part is the last occurrence of a smallest part, n>=1, 1<=k<=n.

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