A006171 -id:A006171 - cp's OEIS frontend

A211857 Number of representations of n as a sum of products of distinct pairs of integers larger than 1, considered to be equivalent when terms or factors are reordered.

1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 2, 0, 3, 1, 3, 2, 5, 1, 7, 3, 8, 5, 11, 4, 16, 9, 17, 12, 25, 13, 34, 20, 37, 28, 53, 32, 69, 46, 78, 63, 108, 71, 136, 100, 160, 134, 210, 152, 265, 211, 313, 268, 403, 316, 506, 421, 596, 528, 759, 629, 943, 814, 1111, 1016

Offset: 0

Alois P. Heinz, Apr 22 2012

nonn

			a(0) = 1: 0 = the empty sum.
a(1) = a(2) = a(3) = 0: no product is < 4.
a(4) = 1: 4 = 2*2.
a(6) = 1: 6 = 2*3.
a(8) = 1: 8 = 2*4.
a(9) = 1: 9 = 3*3.
a(10) = 2: 10 = 2*2 + 2*3 = 2*5.
a(12) = 3: 12 = 2*2 + 2*4 = 2*6 = 3*4.
a(16) = 5: 16 = 2*2 + 2*6 = 2*2 + 3*4 = 2*3 + 2*5 = 2*8 = 4*4.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Cf. A000005, A006171, A038548, A066739, A182269, A182270, A211856.

with(numtheory):
b:= proc(n, i) option remember; local c;
      c:= ceil(tau(i)/2)-1;
      `if`(n=0, 1, `if`(i<2, 0, b(n, i-1)
       +add(b(n-i*j, i-1) *binomial(c, j), j=1..min(c, n/i))))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..70);

b[n_, i_] := b[n, i] = Module[{c}, c = Ceiling[DivisorSigma[0, i]/2]-1; If[n==0, 1, If[i<2, 0, b[n, i-1]+Sum[b[n-i*j, i-1]*Binomial[c, j], {j, 1, Min[c, n/i]}]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Feb 19 2017, translated from Maple *)

G.f.: Product_{k>0} (1+x^k)^(A038548(k)-1). - Vaclav Kotesovec, Aug 19 2019

G.f.: Product_{i>=1} Product_{j=2..i} (1 + x^(i*j)). - Ilya Gutkovskiy, Sep 23 2019

A275585 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_2(k)).

Original entry on oeis.org

1, 1, 6, 16, 52, 128, 373, 913, 2399, 5796, 14298, 33655, 79756, 183078, 419846, 942807, 2106176, 4633208, 10127557, 21870997, 46912648, 99639685, 210206722, 439777198, 914157490, 1886428608, 3869204040, 7884691072, 15976273573, 32182538964, 64484592372, 128518359868, 254868985099, 502950483815, 987904826874, 1931596634076

Offset: 0

Ilya Gutkovskiy, Dec 25 2016

nonn

Euler transform of the sum of squares of divisors (A001157).

Seiichi Manyama, Table of n, a(n) for n = 0..10000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Index entries for sequences related to sums of divisors

Cf. A001157, A027847, A288414.

Product_{k>=1} 1/(1 - x^k)^sigma_m(k): A006171 (m=0), A061256 (m=1), this sequence (m=2), A288391 (m=3), A301542 (m=4), A301543 (m=5), A301544 (m=6), A301545 (m=7), A301546 (m=8), A301547 (m=9).

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*sigma[2](d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 08 2017

nmax = 35; CoefficientList[Series[Product[1/(1 - x^k)^(DivisorSigma[2, k]), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - x^k)^(sigma_2(k)).
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A027847(k)*a(n-k) for n > 0. - Seiichi Manyama, Jun 08 2017
a(n) ~ exp(4*Pi * Zeta(3)^(1/4) * n^(3/4) / (3^(5/4) * 5^(1/4)) - Pi * 5^(1/4) * n^(1/4) / (8 * 3^(7/4) * Zeta(3)^(1/4)) + Zeta(3) / (8*Pi^2)) * Zeta(3)^(1/8) / (2^(3/2) * 15^(1/8) * n^(5/8)). - Vaclav Kotesovec, Mar 23 2018
G.f.: exp(Sum_{k>=1} sigma_3(k)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Oct 26 2018

A356932 Number of multiset partitions of integer partitions of n such that all blocks have odd size.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 24, 42, 74, 130, 224, 383, 653, 1100, 1846, 3079, 5104, 8418, 13827, 22592, 36774, 59613, 96271, 154908, 248441, 397110, 632823, 1005445, 1592962, 2516905, 3966474, 6235107, 9777791, 15297678, 23880160, 37196958, 57819018, 89691934, 138862937

Offset: 0

Gus Wiseman, Sep 11 2022

nonn

			The a(1) = 1 through a(5) = 13 multiset partitions:
  {1}  {2}     {3}        {4}           {5}
       {1}{1}  {111}      {112}         {113}
               {1}{2}     {1}{3}        {122}
               {1}{1}{1}  {2}{2}        {1}{4}
                          {1}{111}      {2}{3}
                          {1}{1}{2}     {11111}
                          {1}{1}{1}{1}  {1}{112}
                                        {2}{111}
                                        {1}{1}{3}
                                        {1}{2}{2}
                                        {1}{1}{111}
                                        {1}{1}{1}{2}
                                        {1}{1}{1}{1}{1}

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

Partitions with odd multiplicities are counted by A055922.
Odd-length multisets are counted by A000302, A027193, A058695, ranked by A026424.
Other types: A050330, A356933, A356934, A356935.
Other conditions: A001970, A006171, A007294, A089259, A107742, A356941.
A000041 counts integer partitions, strict A000009.
A001055 counts factorizations.
Cf. A000670, A011782, A055887, A072233, A117958, A270995, A304969.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[Join@@mps/@IntegerPartitions[n],OddQ[Times@@Length/@#]&]],{n,0,8}]

P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
seq(n) = {my(u=Vec(P(n,1)-P(n,-1))/2); Vec(1/prod(k=1, n, (1 - x^k + O(x*x^n))^u[k])) } \\ Andrew Howroyd, Dec 30 2022

G.f.: 1/Product_{k>=1} (1 - x^k)^A027193(k). - Andrew Howroyd, Dec 30 2022

Terms a(13) and beyond from Andrew Howroyd, Dec 30 2022

A174465 G.f.: exp( Sum_{n>=1} A174466(n)x^n/n ) where A174466(n) = Sum_{d|n} dsigma(n/d)*tau(d).

Original entry on oeis.org

1, 1, 4, 7, 19, 31, 74, 122, 258, 430, 835, 1378, 2557, 4162, 7382, 11932, 20471, 32676, 54634, 86251, 141001, 220371, 353413, 546783, 863043, 1322425, 2057525, 3125092, 4801297, 7230393, 10984924, 16410474, 24679719, 36593278, 54526145, 80272501

Offset: 0

Paul D. Hanna, Apr 04 2010

nonn

Compare to the g.f. of the number of planar partitions of n (A000219):
exp( Sum_{n>=1} sigma_2(n)*x^n/n ) where sigma_2(n) = Sum_{d|n} d*sigma(n/d)*phi(d).
tau(n) = A000005(n) = the number of divisors of n,
and sigma(n) = A000203(n) = sum of divisors of n.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A174466, A000203 (sigma), A000005 (tau), A006171, A007425, A280473, A280487.

nmax = 50; CoefficientList[Series[Product[1/(1-x^(i*j*k)), {i, 1, nmax}, {j, 1, nmax/i}, {k, 1, nmax/i/j}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 04 2017 *)
nmax = 50; A007425 = Table[Sum[DivisorSigma[0, d], {d, Divisors[n]}], {n, 1, nmax}]; s = 1 - x; Do[s *= Sum[Binomial[A007425[[k]], j]*(-1)^j*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; CoefficientList[Series[1/s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 30 2018 *)

{a(n)=polcoeff(exp(sum(m=1,n,x^m/m*sumdiv(m,d,d*sigma(m/d)*sigma(d,0)))+x*O(x^n)),n)}

G.f.: Product_{i>=1, j>=1, k>=1} 1/(1 - x^(i*j*k)). - Vaclav Kotesovec, Jan 04 2017

G.f.: Product_{k>=1} 1/(1 - x^k)^tau_3(k), where tau_3() = A007425. - Ilya Gutkovskiy, May 22 2018

A288391 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_3(k)).

Original entry on oeis.org

1, 1, 10, 38, 156, 534, 2014, 6796, 23312, 76165, 247234, 780343, 2435903, 7453859, 22538336, 67130594, 197666509, 574876417, 1654464954, 4711217687, 13288453688, 37133349758, 102873771662, 282630567325, 770410193747, 2084205092693, 5598070811010

Offset: 0

Seiichi Manyama, Jun 08 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A027848, A288392, A288415.

Product_{k>=1} 1/(1 - x^k)^sigma_m(k): A006171 (m=0), A061256 (m=1), A275585 (m=2), this sequence (m=3).

m:=40; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-q^k)^DivisorSigma(3,k): k in [1..m]]) )); // G. C. Greubel, Oct 30 2018

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*sigma[3](d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 08 2017

nmax = 40; CoefficientList[Series[Product[1/(1-x^k)^DivisorSigma[3, k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 23 2018 *)

m=40; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^sigma(k,3))) \\ G. C. Greubel, Oct 30 2018

a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A027848(k)*a(n-k) for n > 0.
a(n) ~ exp((5*Pi)^(4/5) * Zeta(5)^(1/5) * n^(4/5) / (2^(8/5) * 3^(1/5)) - Zeta'(-3)/2) * Zeta(5)^(121/1200) / ((24*Pi)^(121/1200) * 5^(721/1200) * n^(721/1200)). - Vaclav Kotesovec, Mar 23 2018
G.f.: exp(Sum_{k>=1} sigma_4(k)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Oct 26 2018

A293548 Expansion of Product_{k>=2} 1/(1 - x^k)^omega(k), where omega(k) is the number of distinct primes dividing k (A001221).

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 5, 4, 8, 9, 15, 16, 28, 29, 46, 54, 77, 90, 131, 150, 211, 251, 337, 401, 540, 637, 839, 1006, 1296, 1551, 1995, 2373, 3013, 3610, 4523, 5410, 6754, 8045, 9965, 11897, 14614, 17410, 21313, 25316, 30816, 36615, 44307, 52539, 63387, 74975, 90078

Offset: 0

Ilya Gutkovskiy, Oct 11 2017

nonn

Euler transform of A001221.

N. J. A. Sloane, Transforms

Cf. A001221, A006171, A293549.

nmax = 50; CoefficientList[Series[Product[1/(1 - x^k)^PrimeNu[k], {k, 2, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d PrimeNu[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 50}]

G.f.: Product_{k>=2} 1/(1 - x^k)^b(k), where b(k) = [x^k] Sum_{j>=1} x^prime(j)/(1 - x^prime(j)).

a(0) = 1; a(n) = (1/n)*Sum_{k=1..n} a(n-k)*b(k), b(k) = Sum_{d|k} d*omega(d).

A381993 Number of integer partitions of n that cannot be partitioned into constant multisets with a common sum.

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 4, 13, 13, 25, 33, 54, 54, 99, 124, 166, 207, 295, 352, 488, 591, 780, 987, 1253, 1488, 1951, 2419, 2993, 3665, 4563, 5508, 6840, 8270, 10127, 12289, 14869, 17781, 21635, 25992, 31167, 37184, 44581, 53008, 63259, 75076, 89080, 105531, 124752, 146842, 173516, 204141, 239921, 281461, 329929, 385852

Offset: 0

Gus Wiseman, Mar 17 2025

nonn

			The multiset partition {{2},{2},{1,1},{1,1}} has both properties (constant blocks and common sum), so (2,2,1,1,1,1) is not counted under a(8). We can also use {{2,2},{1,1,1,1}}.
The a(3) = 1 through a(8) = 13 partitions:
  (21)  (31)  (32)    (42)   (43)      (53)
              (41)    (51)   (52)      (62)
              (221)   (321)  (61)      (71)
              (311)   (411)  (322)     (332)
              (2111)         (331)     (431)
                             (421)     (521)
                             (511)     (611)
                             (2221)    (3221)
                             (3211)    (3311)
                             (4111)    (4211)
                             (22111)   (5111)
                             (31111)   (32111)
                             (211111)  (311111)

Twice-partitions of this type (constant with equal) are counted by A279789.
Multiset partitions of this type are ranked by A326534 /\ A355743.
For distinct instead of equal block-sums we have A381717.
These partitions are ranked by A381871, zeros of A381995.
For strict instead of constant blocks we have A381994, see A381719, A382080.
The strict case is A382076.
Normal multiset partitions of this type are counted by A382204.
A001055 counts factorizations, strict A045778.
A050361 counts factorizations into distinct prime powers, see A381715.
A317141 counts coarsenings of prime indices, refinements A300383.
Cf. A000688, A006171, A047966, A265947, A279784, A381453, A381455, A381635, A381636, A381992.

mce[y_]:=Table[ConstantArray[y[[1]],#]&/@ptn,{ptn,IntegerPartitions[Length[y]]}];
Table[Length[Select[IntegerPartitions[n],Length[Select[Join@@@Tuples[mce/@Split[#]],SameQ@@Total/@#&]]==0&]],{n,0,30}]

a(31)-a(54) from Robert Price, Mar 31 2025

A383097 Number of integer partitions of n having more than one permutation with all equal run-sums.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 9, 0, 7, 0, 12, 0, 1, 0, 38, 0, 1, 1, 18, 0, 38, 0, 32, 0, 1, 0, 90, 0, 1, 0, 71, 0, 78, 0, 33, 10, 1, 0, 228, 0, 31, 0, 42, 0, 156, 0, 123, 0, 1, 0, 447, 0, 1, 16, 146, 0, 222, 0, 63, 0, 102, 0, 811, 0, 1, 29, 75, 0, 334, 0

Offset: 0

Gus Wiseman, Apr 17 2025

nonn

			The a(27) = 1 partition is: (9,3,3,3,1,1,1,1,1,1,1,1,1).
The a(4) = 1 through a(16) = 9 partitions (empty columns not shown):
  (211)  (3111)  (422)     (511111)  (633)        (71111111)  (844)
                 (41111)             (6222)                   (82222)
                 (221111)            (33222)                  (442222)
                                     (4221111)                (44221111)
                                     (6111111)                (422221111)
                                     (33111111)               (811111111)
                                     (222111111)              (4411111111)
                                                              (42211111111)
                                                              (222211111111)

These partitions are ranked by A383015, positions of terms > 1 in A382877.
For run-lengths instead of sums we have A383090, ranks A383089, unique A383094.
The complement is A383095 + A383096, ranks A383099 \/ A383100.
For any positive number of permutations we have A383098, ranks A383110.
Counting and ranking partitions by run-lengths and run-sums:
- constant: A047966 (ranks A072774), sums A304442 (ranks A353833)
- distinct: A098859 (ranks A130091), sums A353837 (ranks A353838)
- weakly decreasing: A100882 (ranks A242031), sums A304405 (ranks A357875)
- weakly increasing: A100883 (ranks A304678), sums A304406 (ranks A357861)
- strictly decreasing: A100881 (ranks A304686), sums A304428 (ranks A357862)
- strictly increasing: A100471 (ranks A334965), sums A304430 (ranks A357864)
A275870 counts collapsible partitions, ranks A300273.
A326534 ranks multiset partitions with a common sum, counted by A321455, normal A326518.
A353851 counts compositions with all equal run-sums, ranks A353848.
A382876 counts permutations of prime indices with distinct run-sums, zeros A381636.
Cf. A006171, A329738, A353832, A353839, A353850, A353932, A354584, A381717, A382076.

Table[Length[Select[IntegerPartitions[n],Length[Select[Permutations[#],SameQ@@Total/@Split[#]&]]>1&]],{n,0,15}]

More terms from Bert Dobbelaere, Apr 26 2025

A294363 E.g.f.: exp(Sum_{n>=1} d(n) * x^n), where d(n) is the number of divisors of n.

Original entry on oeis.org

1, 1, 5, 25, 193, 1481, 16021, 167665, 2220065, 30004273, 468585541, 7560838121, 138355144225, 2589359765305, 53501800316693, 1146089983207681, 26457132132638401, 632544682981967585, 16171678558995779845, 426926324177655018553, 11938570457328874969601

Offset: 0

Seiichi Manyama, Oct 29 2017

From Peter Bala, Nov 13 2017: (Start)
The terms of the sequence appear to be of the form 4*m + 1.
It appears that the sequence taken modulo 10 is periodic with period 5. More generally, we conjecture that for k = 2,3,4,... the sequence a(n+k) - a(n) is divisible by k: if true, then for each k the sequence a(n) taken modulo k would be periodic with the exact period dividing k. (End)
From Peter Bala, Mar 28 2022: (Start)
The above conjectures are true. See the Bala link.
a(5*n+2) == 0 (mod 5); a(5*n+3) == 0 (mod 5); a(13*n+9) == 0 (mod 13). (End)

Seiichi Manyama, Table of n, a(n) for n = 0..438
Peter Bala, Integer sequences that become periodic on reduction modulo k for all k

E.g.f.: exp(Sum_{n>=1} sigma_k(n) * x^n): this sequence (k=0), A294361 (k=1), A294362 (k=2).

Cf. A000005, A028342, A295739, A006171, A107742.

nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[0, k]*x^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 05 2018 *)
a[n_] := a[n] = If[n == 0, 1, Sum[k*DivisorSigma[0, k]*a[n-k], {k, 1, n}]/n]; Table[n!*a[n], {n, 0, 20}] (* Vaclav Kotesovec, Sep 06 2018 *)

N=66; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, numdiv(k)*x^k))))

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A000005(k)*a(n-k)/(n-k)! for n > 0.
E.g.f.: Product_{k>=1} exp(x^k/(1 - x^k)). - Ilya Gutkovskiy, Nov 27 2017
Conjecture: log(a(n)/n!) ~ sqrt(2*n*log(n)). - Vaclav Kotesovec, Sep 07 2018

A319647 a(n) = [x^n] Product_{k>=1} 1/(1 - x^k)^sigma_n(k).

Original entry on oeis.org

1, 1, 6, 38, 526, 13074, 702813, 70939556, 13879861574, 5583837482767, 4393101918607162, 6717450870069292051, 21057681806321501744772, 131246096280071506595491449, 1604095619160115980216291007253, 40299198842857238408636666363954678, 2031474817845087309816967328335309651478

Offset: 0

Ilya Gutkovskiy, Oct 26 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..80

Cf. A006171, A061256, A275585, A288391, A301542, A301543, A301544, A301545, A301546, A301547, A321042.

with(numtheory):
b:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*
      sigma[k](d), d=divisors(j))*b(n-j, k), j=1..n)/n)
    end:
a:= n-> b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 26 2018

Table[SeriesCoefficient[Product[1/(1 - x^k)^DivisorSigma[n, k], {k, 1, n}], {x, 0, n}], {n, 0, 16}]
Table[SeriesCoefficient[Product[Product[1/(1 - x^(i j))^(j^n), {j, 1, n}], {i, 1, n}], {x, 0, n}], {n, 0, 16}]
Table[SeriesCoefficient[Exp[Sum[DivisorSigma[n + 1, k] x^k/(k (1 - x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 16}]

{a(n) = polcoeff(prod(k=1, n, 1/(1-x^k+x*O(x^n))^sigma(k, n)), n)} \\ Seiichi Manyama, Oct 27 2018

a(n) = [x^n] Product_{i>=1, j>=1} 1/(1 - x^(i*j))^(j^n).

a(n) = [x^n] exp(Sum_{k>=1} sigma_(n+1)(k)*x^k/(k*(1 - x^k))).

cp's OEIS Frontend

A211857 Number of representations of n as a sum of products of distinct pairs of integers larger than 1, considered to be equivalent when terms or factors are reordered.

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A275585 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_2(k)).

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A356932 Number of multiset partitions of integer partitions of n such that all blocks have odd size.

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A174465 G.f.: exp( Sum_{n>=1} A174466(n)*x^n/n ) where A174466(n) = Sum_{d|n} d*sigma(n/d)*tau(d).

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A288391 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_3(k)).

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A293548 Expansion of Product_{k>=2} 1/(1 - x^k)^omega(k), where omega(k) is the number of distinct primes dividing k (A001221).

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A381993 Number of integer partitions of n that cannot be partitioned into constant multisets with a common sum.

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A383097 Number of integer partitions of n having more than one permutation with all equal run-sums.

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A174465 G.f.: exp( Sum_{n>=1} A174466(n)x^n/n ) where A174466(n) = Sum_{d|n} dsigma(n/d)*tau(d).