A266477 -id:A266477 - cp's OEIS frontend

A265251 Number of partitions of n such that there is exactly one part which occurs three times, while all other parts occur only once.

0, 0, 0, 1, 0, 1, 2, 2, 2, 4, 6, 6, 9, 10, 14, 19, 22, 26, 35, 40, 50, 63, 74, 88, 107, 127, 150, 181, 213, 249, 296, 345, 401, 473, 546, 636, 741, 853, 983, 1138, 1306, 1498, 1722, 1967, 2247, 2574, 2925, 3327, 3788, 4294, 4866, 5516, 6233, 7036, 7947, 8953

Offset: 0

Emeric Deutsch, Dec 28 2015

nonn

Conjecture: a(n) is also the difference between the number of parts in the distinct partitions of n and the number of distinct parts in the odd partitions of n (offset 0). For example, if n = 5, there are 5 parts in the distinct partitions of 5 (5, 41, 32) and 4 distinct parts in the odd partitions of 5 (namely, 5,3,1,1 in 5,311,11111) with difference 1. - George Beck, Apr 22 2017

George E. Andrews has kindly informed me that he has proved this conjecture and the result will be included in his article "Euler's Partition Identity and Two Problems of George Beck" which will appear in The Mathematics Student, 86, Nos. 1-2, January - June (2017). - George Beck, Apr 23 2017

			a(9) = 4 because we have [2,2,2,3], [3,3,3], [1,1,1,2,4], and [1,1,1,6].

Vaclav Kotesovec, Table of n, a(n) for n = 0..15866 (terms 0..10000 from Alois P. Heinz)
George E. Andrews, Euler's Partition Identity and Two Problems of George Beck, The Mathematics Student, 86, 1-2:115-119 (2017); Preprint.
Cristina Ballantine and Richard Bielak, Combinatorial proofs of two Euler type identities due to Andrews, arXiv:1803.06394 [math.CO], 2018.
Cristina Ballantine and Amanda Welch, Beck-type identities for Euler pairs of order $r$, arXiv:2006.02335 [math.NT], 2020.
Cristina Ballantine, Hannah Burson, William Craig, Amanda Folsom, and Boya Wen, Hook length biases and general linear partition inequalities, arXiv:2303.16512 [math.CO], 2023.
Cristina Ballantine and Amanda Welch, Beck-type identities: new combinatorial proofs and a theorem for parts congruent to t mod r, arXiv:2011.08220 [math.CO], 2020.
Cristina Ballantine and Amanda Welch, Beck-type companion identities for Franklin's identity, arXiv:2101.06260 [math.CO], 2021.
Cristina Ballantine and Amanda Welch, Beck-type identities: new combinatorial proofs and a modular refinement, Ramanujan J. (2021).
Cristina Ballantine, Hannah Burson, William Craig, Amanda Folsom, and Boya Wen, Hook length bias in odd versus distinct partitions, Séminaire Lotharingien de Combinatoire, Proc. 35th Conf. Formal Power Series and Alg. Comb. (Davis, 2023) Vol. 89B, Art. #39.
Alexander Berkovich and Aritram Dhar, Finite analogs of partition bias related to hook length two and a variant of Sylvester's map, arXiv:2503.17571 [math.CO], 2025. See p. 19.
Shishuo Fu and Dazhao Tang, Generalizing a partition theorem of Andrews, arXiv:1705.05046 [math.CO], 2017.
Runqiao Li and Andrew Y. Z. Wang, The dual form of Beck type identities, Ramanujan J. (2021).
Aritro Pathak, On certain partition bijections related to Euler's partition problem, arXiv:2004.03596 [math.CO], 2020. Also Discrete Mathematics 345.2 (2022): 112673.

Column k=3 of A266477.

Cf. A015723, A090858, A090867.

g := add(x^(3*k)/(1+x^k), k = 1 .. 100)*mul(1+x^i, i = 1 .. 100): gser := series(g, x = 0, 80): seq(coeff(gser, x, m), m = 0 .. 75);
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n>i*(i+5-4*t)/2, 0,
     `if`(n=0, t, b(n, i-1, t)+`if`(i>n, 0, b(n-i, i-1, t)+
     `if`(t=1 or 3*i>n, 0, b(n-3*i, i-1, 1)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 28 2015

b[n_, i_, t_] := b[n, i, t] = If[n > i*(i + 5 - 4*t)/2, 0, If[n == 0, t, b[n, i - 1, t] + If[i > n, 0, b[n - i, i - 1, t] + If[t == 1 || 3*i > n, 0, b[n - 3*i, i - 1, 1]]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Dec 11 2016, after Alois P. Heinz *)
Take[ CoefficientList[ Expand[ Sum[x^(3k)/(1 + x^k), {k, 60}] Product[1 + x^i, {i, 60}]], x], 60] (* slower than above *) (* Robert G. Wilson v, Apr 24 2017 *)

x='x + O('x^54); concat([0, 0, 0],Vec(sum(k=1, 54, x^(3*k)/(1 + x^k)* prod(i=1, 54, 1 + x^i)))) \\ Indranil Ghosh, Apr 24 2017

G.f.: Sum_{k>=1} x^{3k}/(1+x^k)*Product_{i>=1} (1+x^i).

a(n) ~ c * exp(Pi*sqrt(n/3)) / n^(1/4), where c = 3^(1/4) * (2*log(2) - 1) / (4*Pi) = 0.040456547528... - Vaclav Kotesovec, May 24 2018

A266480 Maximal product of multiplicities of parts of a partition of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 15, 18, 21, 24, 28, 32, 36, 40, 45, 50, 56, 64, 72, 84, 96, 108, 120, 135, 150, 165, 180, 200, 220, 240, 264, 288, 312, 336, 364, 405, 450, 495, 540, 600, 660, 720, 792, 864, 936, 1008, 1092, 1176, 1260, 1365, 1470, 1575

Offset: 0

Emeric Deutsch and Alois P. Heinz, Dec 29 2015

nonn

			a(4) = 4 because the products of the multiplicities of the parts in the partitions [4], [1,3], [2,2], [1,1,2], [1,1,1,1] are 1, 1, 2, 2, 4, respectively.
a(21) = 7*4*2 = 56 for partition [1,1,1,1,1,1,1,2,2,2,2,3,3].

Vaclav Kotesovec, Table of n, a(n) for n = 0..16000 (terms 0..5000 from Alois P. Heinz)

Row lengths of A266477.

Cf. A266871.

b:= proc(n, i) option remember; `if`(n=0 or i=1, max(1, n),
      max(seq(b(n-i*j, i-1)*max(1, j), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..100);

Table[Max@ Map[Times @@ Map[Last, Tally@ #] &, IntegerPartitions@ n], {n, 0, 56}] (* Michael De Vlieger, Dec 31 2015 *)
b[n_, i_] := b[n, i] = If[n==0 || i==1, Max[1, n], Max[Table[b[n-i*j, i-1]*Max[1, j], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Sep 01 2016, after Alois P. Heinz *)

A353698 Number of integer partitions of n whose product equals their length.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 2, 1, 2, 0, 2, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 3, 0, 3, 2, 2, 1, 5, 0, 1, 2, 5, 1, 4, 0, 3, 3, 2, 1, 4, 2, 3, 2, 2, 0, 5, 1, 4, 2, 2, 3, 6, 1, 2, 2, 5, 1, 4, 0, 4, 3, 3, 1, 6, 2, 3, 4, 4, 2, 4, 1, 4, 2, 3, 1, 8, 2, 4, 2, 4, 2, 5, 2, 4, 2

Offset: 0

Gus Wiseman, May 19 2022

nonn

			The a(n) partitions for selected n (A..H = 10..17):
n=9:    n=21:             n=27:                 n=33:
---------------------------------------------------------------------------
51111   B1111111111       E1111111111111        H1111111111111111
321111  72111111111111    921111111111111111    B211111111111111111111
        531111111111111   54111111111111111111  831111111111111111111111
        4221111111111111                        5511111111111111111111111
                                                333111111111111111111111111

Andrew Howroyd, Table of n, a(n) for n = 0..10000

The LHS (product of parts) is counted by A339095, rank statistic A003963.
The RHS (length) is counted by A008284, rank statistic A001222.
These partitions are ranked by A353699.
A266477 counts partitions by product of multiplicities, rank stat A005361.
A353504 counts partitions w/ product less than product of multiplicities.
A353505 counts partitions w/ product greater than product of multiplicities.
A353506 counts partitions w/ prod equal to prod of mults, ranked by A353503.
Cf. A000041, A002033, A098859, A114640, A181819, A225485, A266499, A319000, A325280, A353398, A353507.

Table[Length[Select[IntegerPartitions[n],Times@@#==Length[#]&]],{n,0,30}]

a(r,m=r,p=1,k=0) = {(p==k+r) + sum(m=2, min(m, (k+r)\p),  self()(r-m, min(m,r-m), p*m, k+1))} \\ Andrew Howroyd, Jan 02 2023

Terms a(61) and beyond from Andrew Howroyd, Jan 02 2023

A266871 Number of partitions of n that maximize the product of multiplicities of parts.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1

Offset: 0

Emeric Deutsch and Alois P. Heinz, Jan 04 2016

nonn

			a(8) = 2: [1,1,1,1,1,1,1,1], [1,1,1,1,2,2] (product of multiplicities = 8).
a(9) = 1: [1,1,1,1,1,2,2] (product = 10).
a(10) = 2: [1,1,1,1,1,1,2,2], [1,1,1,1,2,2,2] (product = 12).
a(11) = 1: [1,1,1,1,1,2,2,2] (product = 15).
a(23) = 3: [1,1,1,1,1,1,1,1,1,2,2,2,2,3,3], [1,1,1,1,1,1,1,1,2,2,2,3,3,3], [1,1,1,1,1,1,2,2,2,2,3,3,3] (product = 72).

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Cf. A266477, A266480.

b:= proc(n, i) option remember; local r,l,j;
      if n=0 or i=1 then [max(1, n),1]
    else r:= b(n, i-1);
         for j to iquo(n, i) do
           l:= (w-> [w[1]*j, w[2]])(b(n-i*j, i-1));
           r:= `if`(l[1]>r[1], l,
               `if`(l[1]=r[1], [0, l[2]], 0)+r)
         od; r
      fi
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..120);

b[n_, i_] := b[n, i] = Module[{r, l, j}, If[n == 0 || i == 1, {Max[1, n], 1}, r = b[n, i - 1]; For[j = 1, j <= Quotient[n, i], j++, l = Function[w, {w[[1]]*j, w[[2]]}][b[n - i*j, i - 1]]; r = If[l[[1]] > r[[1]], l, If[l[[1]] == r[[1]], {0, l[[2]]}, 0] + r]]; r]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 120}] (* Jean-François Alcover, Dec 21 2016, translated from Maple *)

a(n) = A266477(n,A266480(n)).

A266325 Smallest integer m such that there is a partition of m with product of multiplicities of parts equal to n.

Original entry on oeis.org

0, 2, 3, 4, 5, 6, 7, 8, 9, 9, 11, 10, 13, 11, 11, 12, 17, 12, 19, 13, 13, 15, 23, 14, 15, 17, 15, 15, 29, 16, 31, 16, 17, 21, 17, 17, 37, 23, 19, 18, 41, 19, 43, 19, 19, 27, 47, 20, 21, 20, 23, 21, 53, 21, 21, 21, 25, 33, 59, 22, 61, 35, 22, 22, 23, 23, 67, 25

Offset: 1

Emeric Deutsch and Alois P. Heinz, Jan 04 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..2500

Cf. A266477.

b:= proc(n, i, p) option remember; `if`(n=0, `if`(p=1, 1, 0),
      `if`(i<1, 0, b(n, i-1, p)+add(`if`(irem(p, j)=0,
       b(n-i*j, i-1, p/j), 0), j=1..n/i)))
    end:
a:= proc(n) option remember; local m;
      if isprime(n) then return n fi;
      for m from 0 do if b(m$2, n)>0 then return m fi od
    end:
seq(a(n), n=1..100);

b[n_, i_, p_] := b[n, i, p] = If[n == 0, If[p == 1, 1, 0], If[i < 1, 0, b[n, i - 1, p] + Sum[If[Mod[p, j] == 0, b[n - i*j, i - 1, p/j], 0], {j, 1, n/i}]]]; a[n_] := a[n] = Module[{m}, If[PrimeQ[n], Return[n]]; For[m = 0, True, m++, If[b[m, m, n] > 0, Return[m]]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Dec 21 2016, translated from Maple *)

a(n) = min { m >= 0 : A266477(m,n) > 0 }.

p in primes => a(p) = p.

A266687 Number of partitions of n with product of multiplicities of parts equal to 4.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 2, 1, 3, 4, 6, 6, 11, 13, 17, 24, 29, 36, 48, 59, 72, 96, 111, 138, 170, 207, 245, 305, 362, 432, 517, 616, 723, 868, 1013, 1194, 1412, 1644, 1915, 2245, 2605, 3019, 3511, 4051, 4677, 5410, 6209, 7125, 8199, 9372, 10718, 12257, 13975, 15902

Offset: 0

Emeric Deutsch and Alois P. Heinz, Jan 02 2016

nonn

			a(6) = 2: [1,1,1,1,2], [1,1,2,2].
a(7) = 1: [1,1,1,1,3].
a(8) = 3: [2,2,2,2], [1,1,3,3], [1,1,1,1,4].
a(9) = 4: [1,2,2,2,2], [1,1,1,1,2,3], [1,1,2,2,3], [1,1,1,1,5].
a(10) = 6: [1,1,2,3,3], [2,2,3,3], [1,1,1,1,2,4], [1,1,2,2,4], [1,1,4,4], [1,1,1,1,6].

Vaclav Kotesovec, Table of n, a(n) for n = 0..18694 (terms 0..10000 from Alois P. Heinz)

Column k=4 of A266477.

b:= proc(n, i, p) option remember; `if`(n=0, `if`(p=1, 1, 0),
      `if`(i<1, 0, b(n, i-1, p)+add(`if`(irem(p, j)=0,
       b(n-i*j, i-1, p/j), 0), j=1..n/i)))
    end:
a:= n-> b(n$2, 4):
seq(a(n), n=0..70);

b[n_, i_, p_] := b[n, i, p] = If[n == 0, If[p == 1, 1, 0], If[i < 1, 0, b[n, i - 1, p] + Sum[If[Mod[p, j] == 0, b[n - i*j, i - 1, p/j], 0], {j, 1, n/i}]]]; a[n_] := b[n, n, 4]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Dec 22 2016, after Alois P. Heinz *)

a(n) ~ c * exp(Pi*sqrt(n/3)) * n^(1/4), where c = 0.0108735520090052... - Vaclav Kotesovec, May 24 2018

A266688 Number of partitions of n with product of multiplicities of parts equal to 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 3, 3, 3, 4, 7, 8, 10, 12, 15, 18, 24, 28, 35, 42, 48, 60, 72, 84, 102, 120, 140, 166, 194, 226, 264, 311, 358, 416, 482, 554, 641, 738, 844, 970, 1112, 1271, 1450, 1654, 1878, 2138, 2429, 2748, 3116, 3524, 3976, 4493, 5065, 5696

Offset: 0

Emeric Deutsch and Alois P. Heinz, Jan 02 2016

nonn

Also the number of partitions of n such that there is exactly one part which occurs 5 times, while all other parts occur only once.

			a(9) = 1: [1,1,1,1,1,4].
a(10) = 3: [2,2,2,2,2], [1,1,1,1,1,2,3], [1,1,1,1,1,5].

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Column k=5 of A266477.

b:= proc(n, i, p) option remember; `if`(i*(p+(i-1)/2)0, 0, (h->
       b(h, min(h, i-1), p/j))(n-i*j)), j=1..min(p, n/i))))
    end:
a:= n-> b(n$2, 5):
seq(a(n), n=0..65);

b[n_, i_, p_] := b[n, i, p] = If[i*(p + (i - 1)/2) < n, 0, If[n == 0, If[p == 1, 1, 0], b[n, i - 1, p] + Sum[If[Mod[p, j] > 0, 0, Function[h, b[h, Min[h, i - 1], p/j]][n - i*j]], {j, 1, Min[p, n/i]}]]];
a[n_] := b[n, n, 5];
Table[a[n], {n, 0, 65}] (* Jean-François Alcover, May 01 2018, translated from Maple *)

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

a(n) ~ c * exp(Pi*sqrt(n/3)) / n^(1/4), where c = (12*log(2) - 7) / (8*3^(3/4)*Pi) = 0.023001573808... - Vaclav Kotesovec, May 24 2018

A266689 Number of partitions of n with product of multiplicities of parts equal to 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 2, 7, 6, 12, 11, 16, 22, 32, 35, 51, 61, 70, 95, 118, 144, 177, 222, 257, 313, 382, 459, 547, 664, 770, 933, 1092, 1275, 1513, 1786, 2070, 2431, 2838, 3287, 3830, 4435, 5094, 5918, 6825, 7821, 9010, 10340, 11820, 13525, 15474

Offset: 0

Emeric Deutsch and Alois P. Heinz, Jan 02 2016

nonn

			a(7) = 1: [1,1,1,2,2].
a(8) = 2: [1,1,1,1,1,1,2], [1,1,2,2,2].
a(11) = 7: [1,1,1,1,1,1,2,3], [1,1,2,2,2,3], [1,1,1,2,3,3], [1,1,3,3,3], [1,1,1,2,2,4], [1,1,1,4,4], [1,1,1,1,1,1,5].

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Column k=6 of A266477.

b:= proc(n, i, p) option remember; `if`(i*(p+(i-1)/2)0, 0, (h->
       b(h, min(h, i-1), p/j))(n-i*j)), j=1..min(p, n/i))))
    end:
a:= n-> b(n$2, 6):
seq(a(n), n=0..65);

b[n_, i_, p_] := b[n, i, p] = If[i*(p + (i - 1)/2) < n, 0, If[n == 0, If[p == 1, 1, 0], b[n, i - 1, p] + Sum[If[Mod[p, j] > 0, 0, Function[h, b[h, Min[h, i - 1], p/j]][n - i*j]], {j, 1, Min[p, n/i]}]]];
a[n_] := b[n, n, 6];
Table[a[n], {n, 0, 65}] (* Jean-François Alcover, May 01 2018, translated from Maple *)

a(n) ~ c * exp(Pi*sqrt(n/3)) * n^(1/4), where c = 0.01368862060... - Vaclav Kotesovec, May 24 2018

A266690 Number of partitions of n with product of multiplicities of parts equal to 7.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 2, 2, 4, 4, 5, 6, 9, 10, 12, 16, 20, 23, 28, 33, 40, 49, 59, 69, 81, 96, 112, 133, 155, 181, 212, 246, 284, 331, 380, 438, 506, 580, 666, 765, 872, 996, 1136, 1294, 1468, 1669, 1894, 2142, 2426, 2740, 3092, 3488, 3926, 4416

Offset: 0

Emeric Deutsch and Alois P. Heinz, Jan 02 2016

nonn

Also the number of partitions of n such that there is exactly one part which occurs 7 times, while all other parts occur only once.

			a(11) = 1: [1,1,1,1,1,1,1,4].
a(12) = 2: [1,1,1,1,1,1,1,2,3], [1,1,1,1,1,1,1,5].

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Column k=7 of A266477.

b:= proc(n, i, p) option remember; `if`(i*(p+(i-1)/2)0, 0, (h->
       b(h, min(h, i-1), p/j))(n-i*j)), j=1..min(p, n/i))))
    end:
a:= n-> b(n$2, 7):
seq(a(n), n=0..65);

b[n_, i_, p_] := b[n, i, p] = If[i*(p + (i - 1)/2) < n, 0, If[n == 0, If[p == 1, 1, 0], b[n, i - 1, p] + Sum[If[Mod[p, j] > 0, 0, Function[h, b[h, Min[h, i - 1], p/j]][n - i*j]], {j, 1, Min[p, n/i]}]]];
a[n_] := b[n, n, 7];
Table[a[n], {n, 0, 65}] (* Jean-François Alcover, May 01 2018, translated from Maple *)

G.f.: Sum_{k>=1} x^(7*k)/(1+x^k) * Product_{j>=1} (1+x^j).

a(n) ~ c * exp(Pi*sqrt(n/3)) / n^(1/4), where c = (60*log(2)-37) / (40*3^(3/4)*Pi) = 0.016019584320... - Vaclav Kotesovec, May 24 2018

A266691 Number of partitions of n with product of multiplicities of parts equal to 8.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 3, 2, 5, 4, 10, 9, 17, 20, 25, 31, 47, 53, 71, 89, 109, 138, 171, 205, 257, 317, 375, 461, 557, 664, 792, 962, 1124, 1352, 1596, 1878, 2215, 2621, 3042, 3584, 4180, 4862, 5658, 6593, 7598, 8826, 10190, 11730, 13516, 15562, 17811

Offset: 0

Emeric Deutsch and Alois P. Heinz, Jan 02 2016

nonn

			a(8) = 2: [1,1,1,1,1,1,1,1], [1,1,1,1,2,2].
a(10) = 3: [1,1,1,1,1,1,1,1,2], [1,1,2,2,2,2], [1,1,1,1,3,3].
a(12) = 5: [1,1,1,1,2,3,3], [1,1,2,2,3,3], [1,1,1,1,1,1,1,1,4], [1,1,1,1,2,2,4], [1,1,1,1,4,4].

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Column k=8 of A266477.

b:= proc(n, i, p) option remember; `if`(i*(p+(i-1)/2)0, 0, (h->
       b(h, min(h, i-1), p/j))(n-i*j)), j=1..min(p, n/i))))
    end:
a:= b(n$2, 8):
seq(a(n), n=0..65);

b[n_, i_, p_] := b[n, i, p] = If[i*(p + (i - 1)/2) < n, 0, If[n == 0, If[p == 1, 1, 0], b[n, i - 1, p] + Sum[If[Mod[p, j] > 0, 0, Function[h, b[h, Min[h, i - 1], p/j]][n - i*j]], {j, 1, Min[p, n/i]}]]];
a[n_] := b[n, n, 8];
Table[a[n], {n, 0, 65}] (* Jean-François Alcover, May 01 2018, translated from Maple *)

a(n) ~ c * exp(Pi*sqrt(n/3)) * n^(3/4), where c = 0.0012263686774... - Vaclav Kotesovec, May 24 2018

cp's OEIS Frontend

A265251 Number of partitions of n such that there is exactly one part which occurs three times, while all other parts occur only once.

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A266480 Maximal product of multiplicities of parts of a partition of n.

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A353698 Number of integer partitions of n whose product equals their length.

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A266871 Number of partitions of n that maximize the product of multiplicities of parts.

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A266325 Smallest integer m such that there is a partition of m with product of multiplicities of parts equal to n.

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A266687 Number of partitions of n with product of multiplicities of parts equal to 4.

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A266688 Number of partitions of n with product of multiplicities of parts equal to 5.

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A266689 Number of partitions of n with product of multiplicities of parts equal to 6.