A295935 -id:A295935 - cp's OEIS frontend

A354911 Number of factorizations of n into relatively prime prime-powers.

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

Offset: 1

Gus Wiseman, Jul 25 2022

nonn

			The a(n) factorizations for n = 6, 12, 24, 36, 48, 72, 96:
  2*3  3*4    3*8      4*9      3*16       8*9        3*32
       2*2*3  2*3*4    2*2*9    2*3*8      2*4*9      3*4*8
              2*2*2*3  3*3*4    3*4*4      3*3*8      2*3*16
                       2*2*3*3  2*2*3*4    2*2*2*9    2*2*3*8
                                2*2*2*2*3  2*3*3*4    2*3*4*4
                                           2*2*2*3*3  2*2*2*3*4
                                                      2*2*2*2*2*3

Wikipedia, Coprime integers.

This is the relatively prime case of A000688, partitions A023894.
Positions of 0's are A246655 (A000961 includes 1).
For strict instead of relatively prime we have A050361, partitions A054685.
Positions of 1's are A000469 (A120944 excludes 1).
For pairwise coprime instead of relatively prime we have A143731.
The version for partitions instead of factorizations is A356067.
A000005 counts divisors.
A001055 counts factorizations.
A001221 counts distinct prime divisors, with sum A001414.
A001222 counts prime-power divisors.
A289509 lists numbers whose prime indices are relatively prime.
A295935 counts twice-factorizations with constant blocks (type PPR).
A355743 lists numbers with prime-power prime indices, squarefree A356065.
Cf. A000837, A023893, A076610, A085970, A106244, A279784, A318721, A355737, A355742, A356064, A356066.

ufacs[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[ufacs[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
Table[Length[Select[ufacs[Select[Divisors[n],PrimePowerQ[#]&],n],GCD@@#<=1&]],{n,100}]

a(n) = A000688(n) if n is nonprime, otherwise a(n) = 0.

A383014 Numbers whose prime indices can be partitioned into constant blocks with a common sum.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 40, 41, 43, 47, 48, 49, 53, 59, 61, 63, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 108, 109, 112, 113, 121, 125, 127, 128, 131, 137, 139, 144, 149, 151, 157, 163, 167, 169

Offset: 1

Gus Wiseman, Apr 22 2025

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.

			The prime indices of 36 are {1,1,2,2}, and a partition into constant blocks with a common sum is: {{2},{2},{1,1}}, so 36 is in the sequence.
The prime indices of 43200 are {1,1,1,1,1,1,2,2,2,3,3}, and a partition into constant blocks with a common sum is: {{{1,1,1,1,1,1},{2,2,2},{3,3}}}, so 43200 is in the sequence.
The prime indices of 520000 are {1,1,1,1,1,1,3,3,3,3,6} and a partition into constant blocks with a common sum is: {{1,1,1,1,1,1},{3,3},{3,3},{6}}, so 520000 is in the sequence.
The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  11: {5}
  12: {1,1,2}
  13: {6}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  29: {10}
  31: {11}
  32: {1,1,1,1,1}
  36: {1,1,2,2}
  37: {12}
  40: {1,1,1,3}

Twice-partitions of this type (constant blocks with a common sum) are counted by A279789.
Includes all elements of A353833.
For distinct sums we have the complement of A381636.
For strict blocks we have the complement of A381719.
For distinct sums and strict blocks we have the complement of A381806.
The complement is A381871, counted by A381993.
These are the positions of positive terms in A381995.
Partitions of this type are counted by A383093.
Constant blocks: A000688, A006171, A279784, A295935, A381453 (lower), A381455 (upper).
A001055 counts factorizations (multiset partitions of prime indices), strict A045778.
A050361 counts factorizations into distinct prime powers.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A317141 counts coarsenings of prime indices, refinements A300383.
Cf. A000720, A000961, A001222, A321469, A381635, A381715, A381716, A381717.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
mce[y_]:=Table[ConstantArray[y[[1]],#]&/@ptn, {ptn,IntegerPartitions[Length[y]]}];
Select[Range[100], Select[Join@@@Tuples[mce/@Split[prix[#]]], SameQ@@Total/@#&]!={}&]

A295920 Number of twice-factorizations of n of type (P,R,R).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Nov 30 2017

nonn

a(n) is also the number of ways to choose a perfect divisor d|n and then a sequence of log_d(n) perfect divisors of d.

			The a(64) = 17 twice-factorizations are:
(2)*(2)*(2)*(2)*(2)*(2)  (2*2)*(2*2)*(2*2)  (2*2*2)*(2*2*2)  (2*2*2*2*2*2)
(2*2)*(2*2)*(4)          (2*2)*(4)*(2*2)    (4)*(2*2)*(2*2)
(2*2)*(4)*(4)            (4)*(2*2)*(4)      (4)*(4)*(2*2)
(2*2*2)*(8)              (8)*(2*2*2)
(4)*(4)*(4)              (4*4*4)
(8)*(8)                  (8*8)
(64)

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A001055, A052409, A052410, A089723, A279789, A281113, A295923, A295924, A295931, A295935.

Table[Sum[Length[Divisors[GCD@@FactorInteger[n^(1/d)][[All,2]]]]^d,{d,Divisors[GCD@@FactorInteger[n][[All,2]]]}],{n,100}]

A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409
A295920(n) = if(1==n,n,my(r); sumdiv(A052409(n), d, if(!ispower(n,d,&r),(1/0),numdiv(A052409(r))^d))); \\ Antti Karttunen, Dec 06 2018, after Mathematica-code

a(n) = Sum_{d|A052409(n)} A000005(A052409(n^(1/d)))^d. - Antti Karttunen, Dec 06 2018, after Mathematica-code

More terms from Antti Karttunen, Dec 06 2018

A296134 Number of twice-factorizations of n of type (R,Q,R).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Dec 05 2017

nonn

a(n) is the number of ways to choose a strict integer partition of a divisor of A052409(n).

			The a(16) = 4 twice-factorizations: (2)*(2*2*2), (2*2*2*2), (4*4), (16).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A000009, A001055, A047966, A047968, A052409, A052410, A089723, A281113, A295923, A295924, A295931, A295935, A296133.

Table[DivisorSum[GCD@@FactorInteger[n][[All,2]],PartitionsQ],{n,100}]

A000009(n,k=(n-!(n%2))) = if(!n,1,my(s=0); while(k >= 1, if(k<=n, s += A000009(n-k,k)); k -= 2); (s));
A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409
A296134(n) = if(1==n,n,sumdiv(A052409(n),d,A000009(d))); \\ Antti Karttunen, Jul 29 2018

From Antti Karttunen, Jul 31 2018: (Start)
a(1) = 1; for n > 1, a(n) = Sum_{d|A052409(n)} A000009(d).
a(n) = A047966(A052409(n)). (End)

More terms from Antti Karttunen, Jul 29 2018

A383093 Number of integer partitions of n that can be partitioned into constant blocks with a common sum.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 7, 2, 9, 5, 9, 2, 23, 2, 11, 10, 24, 2, 33, 2, 36, 12, 15, 2, 87, 7, 17, 17, 53, 2, 96, 2, 79, 16, 21, 14, 196, 2, 23, 18, 154, 2, 166, 2, 99, 54, 27, 2, 431, 9, 85, 22, 128, 2, 303, 18, 261, 24, 33, 2, 771, 2, 35, 73, 331, 20, 422, 2, 198, 28, 216, 2, 1369

Offset: 0

Gus Wiseman, Apr 22 2025

nonn

			The partition (4,4,2,2,2,2,1,1,1,1,1,1,1,1) has two partitions into constant blocks with a common sum: {{4,4},{2,2,2,2},{1,1,1,1,1,1,1,1}} and {{4},{4},{2,2},{2,2},{1,1,1,1},{1,1,1,1}}, so is counted under a(24).
The a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)
                    (211)            (222)                (422)
                    (1111)           (2211)               (2222)
                                     (3111)               (22211)
                                     (21111)              (41111)
                                     (111111)             (221111)
                                                          (2111111)
                                                          (11111111)

Twice-partitions of this type (constant with common) are counted by A279789.
Multiset partitions of this type are ranked by A383309.
The complement is counted by A381993, ranks A381871.
For sets we have the complement of A381994, see A381719, A382080.
Normal multiset partitions of this type are counted by A382203, sets A381718.
For distinct instead of equal block-sums we have A382427.
These partitions are ranked by A383014, nonzeros of A381995.
A000041 counts integer partitions, strict A000009.
A000688 counts factorizations into prime powers, see A381455, A381453.
A001055 counts factorizations, strict A045778, see A317141, A300383, A265947.
A050361 counts factorizations into distinct prime powers, see A381715.
A323774 counts partitions into constant blocks with a common sum
Constant blocks with distinct sums: A381635, A381636, A381717.
Permutation with equal run-sums: A383096, A383098, A383100, A383110
Cf. A006171, A047966, A279784, A295935, A323774, A326534, A353864, A355743, A381716, A382076, A382204.

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}]

Multiset systems of this type have MM-numbers A383309 = A326534 /\ A355743.

Conjecture: We have Sum_{d|n} a(d) = A323774(n), so this is the Moebius transform of A323774.

More terms from Jakub Buczak, May 03 2025

A382301 Number of integer partitions of n having a unique multiset partition into constant blocks with distinct sums.

Original entry on oeis.org

1, 1, 2, 2, 3, 6, 8, 9, 14, 16, 25, 30, 41, 52, 69, 83, 105, 129, 164, 208, 263, 315, 388, 449, 573, 694

Offset: 0

Gus Wiseman, Mar 26 2025

			The a(4) = 3 through a(8) = 14 partitions and their unique multiset partition into constant blocks with distinct sums:
  {4}     {5}       {6}         {7}        {8}
  {22}    {1}{4}    {33}        {1}{6}     {44}
  {1}{3}  {2}{3}    {1}{5}      {2}{5}     {1}{7}
          {11}{3}   {2}{4}      {3}{4}     {2}{6}
          {1}{22}   {11}{4}     {11}{5}    {3}{5}
          {2}{111}  {11}{22}    {1}{33}    {11}{6}
                    {1}{2}{3}   {3}{22}    {2}{33}
                    {1}{11}{3}  {1}{2}{4}  {11}{33}
                                {3}{1111}  {11}{222}
                                           {1}{2}{5}
                                           {1}{3}{4}
                                           {1}{3}{22}
                                           {1}{4}{111}
                                           {1}{111}{22}

For distinct blocks instead of block-sums we have A000726, ranks A004709.
Twice-partitions of this type (constant with distinct) are counted by A279786.
MM-numbers of these multiset partitions are A326535 /\ A355743.
For no choices we have A381717, ranks A381636, zeros of A381635.
The Heinz numbers of these partitions are A381991, positions of 1 in A381635.
Normal multiset partitions of this type are counted by A382203.
For at least one choice we have A382427.
For strict instead of constant blocks we have A382460, ranks A381870.
A000041 counts integer partitions, strict A000009.
A000688 counts factorizations into prime powers, see A381455, A381453.
A001055 counts factorizations, strict A045778, see A317141, A300383, A265947.
A050361 counts factorizations into distinct prime powers.
Cf. A006171, A047966, A279784, A293511, A295935, A353864, A381633, A381716, A381990, A381992, A381993, A382079.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
pfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[pfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
Table[Length[Select[IntegerPartitions[n],Length[Select[pfacs[Times@@Prime/@#],UnsameQ@@hwt/@#&]]==1&]],{n,0,10}]

A382427 Number of integer partitions of n that can be partitioned into constant blocks with distinct sums.

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 11, 14, 19, 28, 39, 50, 70, 91, 120, 161, 203, 260, 338, 426, 556, 695, 863, 1082, 1360, 1685

Offset: 0

Gus Wiseman, Mar 26 2025

Conjecture: Also the number of integer partitions of n having a permutation with all distinct run-sums.

			The partition (3,2,2,2,1) can be partitioned as {{1},{2},{3},{2,2}} or {{1},{3},{2,2,2}}, so is counted under a(10).
The a(1) = 1 through a(7) = 14 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (42)      (52)
                    (1111)  (221)    (51)      (61)
                            (311)    (222)     (322)
                            (2111)   (321)     (331)
                            (11111)  (411)     (421)
                                     (2211)    (511)
                                     (3111)    (2221)
                                     (21111)   (4111)
                                     (111111)  (22111)
                                               (31111)
                                               (211111)
                                               (1111111)

Twice-partitions of this type (constant with distinct) are counted by A279786.
Multiset partitions of this type are ranked by A326535 /\ A355743.
The complement is counted by A381717, ranks A381636, zeros of A381635.
For strict instead of constant blocks we have A381992, ranks A382075.
For a unique choice we have A382301, ranks A381991.
Normal multiset partitions of this type are counted by A382203, sets A381718.
A000041 counts integer partitions, strict A000009.
A000688 counts factorizations into prime powers, see A381455, A381453.
A001055 counts factorizations, strict A045778, see A317141, A300383, A265947.
A050361 counts factorizations into distinct prime powers.
Cf. A006171, A047966, A279784, A295935, A300385, A353864, A381633, A381716, A381990, A381993, A382079, A382876.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
pfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[pfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
Table[Length[Select[IntegerPartitions[n],Select[pfacs[Times@@Prime/@#],UnsameQ@@hwt/@#&]!={}&]],{n,0,10}]

A381807 Number of multisets that can be obtained by choosing a constant partition of each m = 0..n and taking the multiset union.

Original entry on oeis.org

1, 1, 2, 4, 12, 24, 92, 184, 704, 2016, 7600, 15200, 80664, 161328, 601696, 2198824, 9868544, 19737088, 102010480, 204020960

Offset: 0

Gus Wiseman, Mar 13 2025

A constant partition is a multiset whose parts are all equal. There are A000005(n) constant partitions of n.

			The a(1) = 1 through a(4) = 12 multisets:
  {1}  {1,2}    {1,2,3}        {1,2,3,4}
       {1,1,1}  {1,1,1,3}      {1,1,1,3,4}
                {1,1,1,1,2}    {1,2,2,2,3}
                {1,1,1,1,1,1}  {1,1,1,1,2,4}
                               {1,1,1,2,2,3}
                               {1,1,1,1,1,1,4}
                               {1,1,1,1,1,2,3}
                               {1,1,1,1,2,2,2}
                               {1,1,1,1,1,1,1,3}
                               {1,1,1,1,1,1,2,2}
                               {1,1,1,1,1,1,1,1,2}
                               {1,1,1,1,1,1,1,1,1,1}

The number of possible choices was A066843.
Multiset partitions into constant blocks: A006171, A279784, A295935.
Choosing prime factors: A355746, A355537, A327486, A355744, A355742, A355741.
Choosing divisors: A355747, A355733.
Sets of constant multisets with distinct sums: A381635, A381636, A381716.
Strict instead of constant partitions: A381808, A058694, A152827.
A000041 counts integer partitions, strict A000009, constant A000005.
A000688 counts multiset partitions into constant blocks.
A050361 and A381715 count multiset partitions into constant multisets.
A066723 counts partitions coarser than {1..n}, primorial case of A317141.
A265947 counts refinement-ordered pairs of integer partitions.
A321470 counts partitions finer than {1..n}, primorial case of A300383.
Cf. A001970, A018818, A213385, A299200, A321467, A321468, A321471, A321514, A355731, A381453, A381455.

Table[Length[Union[Sort/@Join@@@Tuples[Select[IntegerPartitions[#],SameQ@@#&]&/@Range[n]]]],{n,0,10}]

Primorial case of A381453: a(n) = A381453(A002110(n)).

a(16)-a(19) from Christian Sievers, Jun 04 2025

A301762 Number of ways to choose a constant rooted partition of each part in a rooted partition of n.

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 21, 34, 55, 90, 143, 220, 347, 528, 805, 1226, 1831, 2719, 4048, 5940, 8710, 12714, 18403, 26529, 38220, 54679, 77899, 110810, 156848, 221181, 311635, 436705, 610597, 852125, 1184928, 1644136, 2276551, 3142523, 4328960, 5953523, 8167209

Offset: 1

Gus Wiseman, Mar 26 2018

nonn

A rooted partition of n is an integer partition of n - 1.

			The a(5) = 7 rooted twice-partitions where the latter rooted partitions are constant: (3), (111), (2)(), (11)(), (1)(1), (1)()(), ()()()().

Cf. A002865, A063834, A093637, A279784, A295935, A300383, A301422, A301462, A301467, A301480, A301706.

Table[Sum[Product[If[k===1,1,DivisorSigma[0,k-1]],{k,ptn}],{ptn,IntegerPartitions[n-1]}],{n,20}]

O.g.f.: Product_{n>0} 1/(1 - d(n-1) x^n) where d(n) = A000005(n) and d(0) = 1.

A357859 Number of integer factorizations of 2n into distinct even factors.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 1, 3, 1, 3, 1, 4, 1, 2, 1, 4, 1, 2, 1, 5, 1, 3, 1, 3, 1, 2, 1, 7, 1, 2, 1, 3, 1, 3, 1, 5, 1, 2, 1, 6, 1, 2, 1, 5, 1, 3, 1, 3, 1, 3, 1, 7, 1, 2, 1, 3, 1, 3, 1, 7, 1, 2, 1, 6, 1, 2, 1

Offset: 1

Gus Wiseman, Oct 17 2022

nonn

			The a(n) factorizations for n = 2, 4, 12, 24, 32, 48, 60, 96:
  (4)  (8)    (24)    (48)     (64)     (96)      (120)     (192)
       (2*4)  (4*6)   (6*8)    (2*32)   (2*48)    (2*60)    (2*96)
              (2*12)  (2*24)   (4*16)   (4*24)    (4*30)    (4*48)
                      (4*12)   (2*4*8)  (6*16)    (6*20)    (6*32)
                      (2*4*6)           (8*12)    (10*12)   (8*24)
                                        (2*6*8)   (2*6*10)  (12*16)
                                        (2*4*12)            (4*6*8)
                                                            (2*4*24)
                                                            (2*6*16)
                                                            (2*8*12)

The version for partitions instead of factorizations is A000009.
Positions of 1's are A004280.
The non-strict version is A340785.
Including odd n gives A357860.
A000005 counts divisors.
A001055 counts factorizations.
A001221 counts distinct prime factors, sum A001414.
A001222 counts prime-power divisors.
A050361 counts strict factorizations into prime powers.
Cf. A000688, A000961, A023894, A295935, A318721.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[2*n],UnsameQ@@#&&OddQ[Times@@(#+1)]&]],{n,100}]

cp's OEIS Frontend

A354911 Number of factorizations of n into relatively prime prime-powers.

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A383014 Numbers whose prime indices can be partitioned into constant blocks with a common sum.

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A295920 Number of twice-factorizations of n of type (P,R,R).

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A296134 Number of twice-factorizations of n of type (R,Q,R).

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A383093 Number of integer partitions of n that can be partitioned into constant blocks with a common sum.

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A382301 Number of integer partitions of n having a unique multiset partition into constant blocks with distinct sums.

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A382427 Number of integer partitions of n that can be partitioned into constant blocks with distinct sums.

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A381807 Number of multisets that can be obtained by choosing a constant partition of each m = 0..n and taking the multiset union.

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