A327908 -id:A327908 - cp's OEIS frontend

A358833 Number of rectangular twice-partitions of n of type (P,R,P).

1, 1, 3, 4, 8, 8, 17, 16, 32, 34, 56, 57, 119, 102, 179, 199, 335, 298, 598, 491, 960, 925, 1441, 1256, 2966, 2026, 3726, 3800, 6488, 4566, 11726, 6843, 16176, 14109, 21824, 16688, 49507, 21638, 50286, 50394, 99408, 44584, 165129, 63262, 208853, 205109, 248150

Offset: 0

Gus Wiseman, Dec 04 2022

nonn

A twice-partition of n is a sequence of integer partitions, one of each part of an integer partition of n, so these are twice-partitions of n into partitions with constant lengths and constant sums.

			The a(1) = 1 through a(5) = 8 twice-partitions:
  (1)  (2)     (3)        (4)           (5)
       (11)    (21)       (22)          (32)
       (1)(1)  (111)      (31)          (41)
               (1)(1)(1)  (211)         (221)
                          (1111)        (311)
                          (2)(2)        (2111)
                          (11)(11)      (11111)
                          (1)(1)(1)(1)  (1)(1)(1)(1)(1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Gus Wiseman, Sequences enumerating triangles of integer partitions

This is the rectangular case of A279787.
This is the case of A306319 with constant sums.
For distinct instead of constant lengths and sums we have A358832.
The version for multiset partitions of integer partitions is A358835.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
A281145 counts same-trees.
Cf. A000041, A000219, A001970, A008284, A141199, A327908, A358823, A358831.

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],SameQ@@Length/@#&&SameQ@@Total/@#&]],{n,0,10}]

P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
seq(n) = {my(u=Vec(P(n,y)-1)); concat([1], vector(n, n, sumdiv(n, d, my(p=u[n/d]); sum(j=1, n/d, polcoef(p, j, y)^d))))} \\ Andrew Howroyd, Dec 31 2022

a(n) = Sum_{d|n} Sum_{j=1..n/d} A008284(n/d, j)^d for n > 0. - Andrew Howroyd, Dec 31 2022

Terms a(21) and beyond from Andrew Howroyd, Dec 31 2022

A358835 Number of multiset partitions of integer partitions of n with constant block sizes and constant block sums.

Original entry on oeis.org

1, 1, 3, 4, 8, 8, 17, 16, 31, 34, 54, 57, 108, 102, 166, 191, 294, 298, 504, 491, 803, 843, 1251, 1256, 2167, 1974, 3133, 3226, 4972, 4566, 8018, 6843, 11657, 11044, 17217, 15010, 28422, 21638, 38397, 35067, 58508, 44584, 91870, 63262, 125114, 106264, 177483

Offset: 0

Gus Wiseman, Dec 05 2022

nonn

			The a(1) = 1 through a(6) = 17 multiset partitions:
  {1}  {2}     {3}        {4}           {5}              {6}
       {11}    {12}       {13}          {14}             {15}
       {1}{1}  {111}      {22}          {23}             {24}
               {1}{1}{1}  {112}         {113}            {33}
                          {1111}        {122}            {114}
                          {2}{2}        {1112}           {123}
                          {11}{11}      {11111}          {222}
                          {1}{1}{1}{1}  {1}{1}{1}{1}{1}  {1113}
                                                         {1122}
                                                         {3}{3}
                                                         {11112}
                                                         {111111}
                                                         {12}{12}
                                                         {2}{2}{2}
                                                         {111}{111}
                                                         {11}{11}{11}
                                                         {1}{1}{1}{1}{1}{1}

Andrew Howroyd, Table of n, a(n) for n = 0..1000

For just constant sums we have A305551, ranked by A326534.
For just constant lengths we have A319066, ranked by A320324.
The version for set partitions is A327899.
For distinct instead of constant lengths and sums we have A358832.
The version for twice-partitions is A358833.
A001970 counts multiset partitions of integer partitions.
A063834 counts twice-partitions, strict A296122.
Cf. A000219, A007425, A141199, A327908, A356932, A358831.

Table[If[n==0,1,Length[Union[Sort/@Join@@Table[Select[Tuples[IntegerPartitions[d],n/d],SameQ@@Length/@#&],{d,Divisors[n]}]]]],{n,0,20}]

P(n,y) = 1/prod(k=1, n, 1 - y*x^k + O(x*x^n))
seq(n) = {my(u=Vec(P(n,y)-1)); concat([1], vector(n, n, sumdiv(n, d, my(p=u[n/d]); sum(j=1, n/d, binomial(d + polcoef(p, j, y) - 1, d)))))} \\ Andrew Howroyd, Dec 31 2022

a(n) = Sum_{d|n} Sum_{j=1..n/d} binomial(d + A008284(n/d, j) - 1, d) for n > 0. - Andrew Howroyd, Dec 31 2022

Terms a(41) and beyond from Andrew Howroyd, Dec 31 2022

A327899 Number of set partitions of {1..n} with equal block sizes and equal block sums.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 6, 3, 2, 1, 63, 1, 2, 317, 657, 1, 4333, 1, 9609

Offset: 0

Gus Wiseman, Sep 29 2019

			The a(8) = 6 set partitions:
     {{1,2,3,4,5,6,7,8}}
    {{1,2,7,8},{3,4,5,6}}
    {{1,3,6,8},{2,4,5,7}}
    {{1,4,5,8},{2,3,6,7}}
    {{1,4,6,7},{2,3,5,8}}
  {{1,8},{2,7},{3,6},{4,5}}

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Set partitions with equal block-sizes are A038041.
Set partitions with equal block-sums are A035470.
Cf. A000110, A000258, A005651, A007837, A008277, A275780, A300335, A319189, A326512, A326513, A326515, A327908.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],And[SameQ@@Length/@#,SameQ@@Total/@#]&]],{n,0,8}]

A327900 Nonprime squarefree numbers whose prime indices all have the same Omega (number of prime factors counted with multiplicity).

Original entry on oeis.org

1, 15, 33, 51, 55, 85, 91, 93, 123, 155, 161, 165, 177, 187, 201, 203, 205, 249, 255, 295, 299, 301, 327, 329, 335, 341, 377, 381, 415, 451, 465, 471, 511, 527, 537, 545, 553, 559, 561, 573, 611, 615, 633, 635, 649, 667, 679, 697, 703, 707, 723, 737, 785, 831

Offset: 1

Gus Wiseman, Sep 30 2019

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.

			The sequence of terms together with their prime indices begins:
    1: {}
   15: {2,3}
   33: {2,5}
   51: {2,7}
   55: {3,5}
   85: {3,7}
   91: {4,6}
   93: {2,11}
  123: {2,13}
  155: {3,11}
  161: {4,9}
  165: {2,3,5}
  177: {2,17}
  187: {5,7}
  201: {2,19}
  203: {4,10}
  205: {3,13}
  249: {2,23}
  255: {2,3,7}
  295: {3,17}

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

The case including primes and nonsquarefree numbers is A320324.
The version for sum of prime indices is A327901.
The version for mean of prime indices is A327902.
Cf. A001222, A038041, A056239, A078175, A112798, A306017, A306021, A316413, A317583, A322794, A327908.

Select[Range[1000],!PrimeQ[#]&&SquareFreeQ[#]&&SameQ@@PrimeOmega/@PrimePi/@First/@FactorInteger[#]&]

A327901 Nonprime squarefree numbers whose prime indices all have the same sum of prime indices (A056239).

Original entry on oeis.org

1, 35, 143, 209, 247, 391, 493, 629, 667, 851, 901, 1073, 1219, 1333, 1457, 1537, 1891, 1961, 2021, 2201, 2623, 2717, 2759, 2867, 2993, 3053, 3239, 3337, 3827, 3977, 4061, 4183, 4223, 4331, 4387, 4633, 5429, 5633, 5767, 5959, 6157, 6191, 6319, 7081, 7093, 7519

Offset: 1

Gus Wiseman, Sep 30 2019

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.

			The sequence of terms together with their prime indices begins:
     1: {}
    35: {3,4}
   143: {5,6}
   209: {5,8}
   247: {6,8}
   391: {7,9}
   493: {7,10}
   629: {7,12}
   667: {9,10}
   851: {9,12}
   901: {7,16}
  1073: {10,12}
  1219: {9,16}
  1333: {11,14}
  1457: {11,15}
  1537: {10,16}
  1891: {11,18}
  1961: {12,16}
  2021: {14,15}
  2201: {11,20}

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

The version including primes and nonsquarefree numbers is A326534.
The version for number of prime indices is A327900.
The version for mean of prime indices is A327902.
Cf. A001222, A005117, A056239, A112798, A302242, A306021, A320324, A326566, A326574, A327908.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],!PrimeQ[#]&&SquareFreeQ[#]&&SameQ@@Total/@primeMS/@primeMS[#]&];

A327902 Nonprime squarefree numbers whose prime indices all have the same average of prime indices (A326567/A326568).

Original entry on oeis.org

1, 21, 57, 115, 133, 145, 159, 371, 393, 399, 515, 535, 565, 667, 803, 869, 917, 933, 1007, 1067, 1113, 1963, 2021, 2095, 2157, 2165, 2177, 2249, 2285, 2315, 2363, 2369, 2461, 2489, 2599, 2705, 2751, 2839, 2987, 3021, 3103, 3277, 3335, 3707, 3859, 4331, 4367

Offset: 1

Gus Wiseman, Sep 30 2019

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.

			The sequence of terms together with their prime indices begins:
     1: {}
    21: {2,4}
    57: {2,8}
   115: {3,9}
   133: {4,8}
   145: {3,10}
   159: {2,16}
   371: {4,16}
   393: {2,32}
   399: {2,4,8}
   515: {3,27}
   535: {3,28}
   565: {3,30}
   667: {9,10}
   803: {5,21}
   869: {5,22}
   917: {4,32}
   933: {2,64}
  1007: {8,16}
  1067: {5,25}

The version including primes and nonsquarefree numbers is A326536.
The version for number of prime indices is A327900.
The version for sum of prime indices is A327901.
Cf. A001222, A005117, A056239, A078175, A102627, A112798, A316413, A326567/A326568, A326621, A327908.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],!PrimeQ[#]&&SquareFreeQ[#]&&SameQ@@Mean/@primeMS/@primeMS[#]&];

cp's OEIS Frontend

A358833 Number of rectangular twice-partitions of n of type (P,R,P).

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A358835 Number of multiset partitions of integer partitions of n with constant block sizes and constant block sums.

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A327899 Number of set partitions of {1..n} with equal block sizes and equal block sums.

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A327900 Nonprime squarefree numbers whose prime indices all have the same Omega (number of prime factors counted with multiplicity).

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A327901 Nonprime squarefree numbers whose prime indices all have the same sum of prime indices (A056239).

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A327902 Nonprime squarefree numbers whose prime indices all have the same average of prime indices (A326567/A326568).

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