A027750 -id:A027750 - cp's OEIS frontend

A378313 Numbers m such that such that Sum_{i=1..k} A027750(m,i)^i = m for some k <= A000005(m).

1, 130, 135, 288, 5083, 8064, 10130, 374057639685, 3138436947541900183, 5386775810449231243, 74220449392444960903, 153525475816743446263, 1388286039882808958923, 8020029492466254993943, 8756593744534084572523, 16468366959402667137403

Offset: 1

Max Alekseyev, Nov 22 2024

There are many terms of the form 1 + p^2 + q^3 with primes p < q. Next known term not of this form is 1254382690393861635950014154836028.

Cf. A001248, A027750, A064510, A180851, A190940, A194269, A307137, A378314.

A132587 Let b(k) be the k-th term of the flattened irregular array where the m-th row contains the positive divisors of m. (b(k) = A027750(k).) Let c(k) be the k-th term of the flattened irregular array where the m-th row contains the positive integers that are <= m and are coprime to m. (c(k) = A038566(k).) Then a(n) = gcd(b(n),c(n)).

Original entry on oeis.org

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

Offset: 1

Leroy Quet, Aug 23 2007

nonn

			A027750: 1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 2, 3, 6, ...
A038566: 1, 1, 1, 2, 1, 3, 1, 2, 3, 4, 1, 5, 1, 2, ...
The 14th terms of each list are 6 and 2.
So a(14) = gcd(6,2) = 2.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A132587

Cf. A132588, A132589, A027750, A038566.

PARI
```
See Links section.
```

More terms from Rémy Sigrist, Feb 07 2019

A132588 Let b(k) be the k-th term of the flattened irregular array where the m-th row contains the positive divisors of m (b(k) = A027750(k)). Then a(n) = gcd(b(n), n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 4, 1, 5, 1, 2, 1, 2, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 15, 1, 1, 4, 1, 2, 1, 1, 1, 2, 1, 2, 3, 2, 1, 1, 1, 2, 1, 1, 5, 2, 1, 1, 1, 7, 1, 2, 1, 2, 1, 1, 1, 2, 1, 4, 3, 2, 1, 12, 1, 1

Offset: 1

Leroy Quet, Aug 23 2007

nonn

			A027750: 1,1,2,1,3,1,2,4,1,5,1,2,3,6,...
The 14th term of this list is 6.
So a(14) = GCD(6,14) = 2.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A132587, A132589, A027750.

for (m=1, oo, fordiv (m, d, print1 (gcd(d, n++) ", "); if (n==86, break (2)))) \\ Rémy Sigrist, Feb 06 2019

More terms from Rémy Sigrist, Feb 06 2019

A292226 Composite numbers m (in increasing order) for which the m-th row polynomial of A027750 in rising powers is irreducible over the integers.

Original entry on oeis.org

4, 9, 12, 16, 24, 25, 30, 36, 40, 45, 48, 49, 56, 60, 63, 64, 70, 72, 80, 81, 84, 90, 96, 105, 108, 112, 120, 121, 126, 132, 135, 140, 144, 150, 154, 160, 165, 168, 169, 175, 176, 180, 182, 189, 192, 195, 198, 200, 208, 210, 216, 220, 224, 225, 231, 234, 240, 252, 260, 264, 270, 273, 275, 280, 286, 288, 289, 297, 300

Offset: 1

Wolfdieter Lang, Oct 29 2017

nonn

The considered integer polynomials of degree A032741(a(n)) are P(a(n), x) = Sum_{k=0..A032741(a(n))} A027750(a(n), k+1)*x^k for n >= 1.
P(1, x) = 1 (constant) and P(prime(n), x) = 1 + prime(n)*x are trivial.
The other polynomials corresponding to composite numbers from A002808 but not in the present sequence factorize into integer polynomials.
This entry was motivated by the proposal A291127 by Michel Lagneau giving the numbers m for which P(m, x) = Sum_{k=0..A032741(m)} A027750(m, k+1)*x^k has at least two purely imaginary zeros. The present composite a(n) numbers do not appear in A291127. Other composite numbers also do not appear, like 18, 20, 28, 32, 44, ...
From Robert Israel, Oct 31 2017: (Start)
Contains p^(q-1) if p is prime and q is an odd prime.
Disjoint from A006881. (End)

			n = 1: P(4, x) = 1 + 2*x + 4*x^2 of degree A032741(4) = 2.
The composite number 6 is not a member of this sequence because P(6, x) = 1 + 2*x + 3*x^2 + 6*x^3 of degree A032741(6) = 3 factorizes as (1 + 2*x)*(1 + 3*x^2).
m = 18 is not a member of the sequence because P(18, x) = 1 + 2*x + 3*x^2 + 6*x^3 + 9*x^4 + 18*x^5 = (1 + 2*x)*(1 + 3*x^2 + 9*x^4). m = 18 does also not appear in A291127.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A006881, A027750, A032741, A291127.

filter:= proc(n) local d,i,x;
  if isprime(n) then return false fi;
  d:= numtheory:-divisors(n);
  irreduc(add(d[i]*x^(i-1),i=1..nops(d)))
end proc:
select(filter, [$2..1000]); # Robert Israel, Oct 31 2017

P[n_, x_] := (d = Divisors[n]).x^Range[0, Length[d] - 1];
okQ[n_] := CompositeQ[n] && IrreduciblePolynomialQ[P[n, x]];
Select[Range[300], okQ] (* Jean-François Alcover, Oct 30 2017 *)

A321725 Irregular triangle read by rows where T(n,k) is the number of d X d non-normal semi-magic squares with d = A027750(n,k) and sum of all entries equal to n.

Original entry on oeis.org

1, 1, 2, 1, 6, 1, 3, 24, 1, 120, 1, 4, 21, 720, 1, 5040, 1, 5, 282, 40320, 1, 55, 362880, 1, 6, 6210, 3628800, 1, 39916800, 1, 7, 120, 2008, 202410, 479001600, 1, 6227020800, 1, 8, 9135630, 87178291200, 1, 231, 153040, 1307674368000, 1, 9, 10147

Offset: 1

Gus Wiseman, Nov 18 2018

A non-normal semi-magic square is a nonnegative integer square matrix with all row sums and column sums equal to d, for some d|n.

			Triangle begins:
   1
   1   2
   1   6
   1   3  24
   1 120
   1   4  21 720
The T(6,2) = 4 semi-magic squares (zeros not shown):
  [3  ] [2 1] [1 2] [  3]
  [  3] [1 2] [2 1] [3  ]
The T(6,3) = 21 semi-magic squares (zeros not shown):
  [2    ] [2    ] [2    ] [1 1  ] [1 1  ] [1 1  ] [1 1  ]
  [  2  ] [  1 1] [    2] [1 1  ] [1   1] [  1 1] [    2]
  [    2] [  1 1] [  2  ] [    2] [  1 1] [1   1] [1 1  ]
.
  [1   1] [1   1] [1   1] [1   1] [  2  ] [  2  ] [  2  ]
  [1 1  ] [1   1] [  2  ] [  1 1] [2    ] [1   1] [    2]
  [  1 1] [  2  ] [1   1] [1 1  ] [    2] [1   1] [2    ]
.
  [  1 1] [  1 1] [  1 1] [  1 1] [    2] [    2] [    2]
  [2    ] [1 1  ] [1   1] [  1 1] [2    ] [1 1  ] [  2  ]
  [  1 1] [1   1] [1 1  ] [2    ] [  2  ] [1 1  ] [2    ]

Chai Wah Wu, Table of n, a(n) for n = 1..60
Wikipedia, Magic square
Index entries for sequences related to magic squares

Cf. A006052, A007016, A027750, A120732, A319056, A319616.

Cf. A321718, A321719, A321721, A321722, A321724.

prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
Table[Length[Select[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[k]==Union[Last/@#],SameQ@@Total/@prs2mat[#],SameQ@@Total/@Transpose[prs2mat[#]]]&]],{n,5},{k,Divisors[n]}]

T(n, A000005(n)) = n!. Sum_k T(n,k) = A321719(n). - Chai Wah Wu, Jan 15 2019

a(15)-a(48) from Chai Wah Wu, Jan 15 2019

Edited by Peter Munn, Mar 05 2025

A322786 Irregular triangle read by rows where T(n,k) is the number of multiset partitions of a multiset with d = A027750(n,k) copies of each integer from 1 to n/d.

Original entry on oeis.org

1, 2, 2, 5, 3, 15, 9, 5, 52, 7, 203, 66, 31, 11, 877, 15, 4140, 712, 109, 22, 21147, 686, 30, 115975, 10457, 339, 42, 678570, 56, 4213597, 198091, 27036, 6721, 1043, 77, 27644437, 101, 190899322, 4659138, 2998, 135, 1382958545, 1688360, 58616, 176

Offset: 1

Gus Wiseman, Dec 26 2018

			Triangle begins:
        1
        2       2
        5       3
       15       9       5
       52       7
      203      66      31      11
      877      15
     4140     712     109      22
    21147     686      30
   115975   10457     339      42
   678570      56
  4213597  198091   27036    6721    1043      77
For example, row 4 counts the following multiset partitions.
  {{1,2,3,4}}        {{1,1,2,2}}        {{1,1,1,1}}
  {{1},{2,3,4}}      {{1},{1,2,2}}      {{1},{1,1,1}}
  {{1,2},{3,4}}      {{1,1},{2,2}}      {{1,1},{1,1}}
  {{1,3},{2,4}}      {{1,2},{1,2}}      {{1},{1},{1,1}}
  {{1,4},{2,3}}      {{2},{1,1,2}}      {{1},{1},{1},{1}}
  {{2},{1,3,4}}      {{1},{1},{2,2}}
  {{3},{1,2,4}}      {{1},{2},{1,2}}
  {{4},{1,2,3}}      {{2},{2},{1,1}}
  {{1},{2},{3,4}}    {{1},{1},{2},{2}}
  {{1},{3},{2,4}}
  {{1},{4},{2,3}}
  {{2},{3},{1,4}}
  {{2},{4},{1,3}}
  {{3},{4},{1,2}}
  {{1},{2},{3},{4}}

Andrew Howroyd, Table of n, a(n) for n = 1..207 (first 50 rows)

Row sums are A322784. First column is A000110.

Cf. A001055, A005176, A027750, A056239, A072774, A100778, A219727, A295193, A306017, A319190, A319612, A322784, A322785, A322787, A322788, A322792.

u[n_,k_]:=u[n,k]=If[n==1,1,Sum[u[n/d,d],{d,Select[Rest[Divisors[n]],#<=k&]}]];
Table[Table[u[Array[Prime,n/d,1,Times]^d,Array[Prime,n/d,1,Times]^d],{d,Divisors[n]}],{n,10}]

\\ needs T(n,k) from A219727.
Row(n)={[T(d,n/d) | d<-divisors(n)]}
{ for(n=1, 12, print(Row(n))) } \\ Andrew Howroyd, Jan 11 2020

T(n,k) = A001055(A002110(n/d)^d), where d = A027750(n,k).

T(n,k) = A219727(d, n/d), where d = A027750(n, k). - Andrew Howroyd, Jan 11 2020

Edited by Peter Munn, Mar 05 2025

A322787 Irregular triangle read by rows where T(n,k) is the number of non-isomorphic multiset partitions of a multiset with d = A027750(n, k) copies of each integer from 1 to n/d.

Original entry on oeis.org

1, 2, 2, 3, 3, 5, 7, 5, 7, 7, 11, 23, 21, 11, 15, 15, 22, 79, 66, 22, 30, 162, 30, 42, 274, 192, 42, 56, 56, 77, 1003, 1636, 1338, 565, 77, 101, 101, 135, 3763, 1579, 135, 176, 19977, 10585, 176, 231, 14723, 43686, 4348, 231, 297, 297, 385, 59663, 298416, 82694, 11582, 385

Offset: 1

Gus Wiseman, Dec 26 2018

			Triangle begins:
   1
   2   2
   3   3
   5   7   5
   7   7
  11  23  21  11
  15  15
  22  79  66  22
  30 162  30
  42 274 192  42
Non-isomorphic representatives of the multiset partitions counted under row 6:
{123456}           {112233}           {111222}           {111111}
{1}{23456}         {1}{12233}         {1}{11222}         {1}{11111}
{12}{3456}         {11}{2233}         {11}{1222}         {11}{1111}
{123}{456}         {112}{233}         {111}{222}         {111}{111}
{1}{2}{3456}       {12}{1233}         {112}{122}         {1}{1}{1111}
{1}{23}{456}       {123}{123}         {12}{1122}         {1}{11}{111}
{12}{34}{56}       {1}{1}{2233}       {1}{1}{1222}       {11}{11}{11}
{1}{2}{3}{456}     {1}{12}{233}       {1}{11}{222}       {1}{1}{1}{111}
{1}{2}{34}{56}     {11}{22}{33}       {11}{12}{22}       {1}{1}{11}{11}
{1}{2}{3}{4}{56}   {11}{23}{23}       {1}{12}{122}       {1}{1}{1}{1}{11}
{1}{2}{3}{4}{5}{6} {1}{2}{1233}       {1}{2}{1122}       {1}{1}{1}{1}{1}{1}
                   {12}{13}{23}       {12}{12}{12}
                   {1}{23}{123}       {2}{11}{122}
                   {2}{11}{233}       {1}{1}{1}{222}
                   {1}{1}{2}{233}     {1}{1}{12}{22}
                   {1}{1}{22}{33}     {1}{1}{2}{122}
                   {1}{1}{23}{23}     {1}{2}{11}{22}
                   {1}{2}{12}{33}     {1}{2}{12}{12}
                   {1}{2}{13}{23}     {1}{1}{1}{2}{22}
                   {1}{2}{3}{123}     {1}{1}{2}{2}{12}
                   {1}{1}{2}{2}{33}   {1}{1}{1}{2}{2}{2}
                   {1}{1}{2}{3}{23}
                   {1}{1}{2}{2}{3}{3}

Andrew Howroyd, Table of n, a(n) for n = 1..207 (rows 1..50)

Row sums are A306017. First column is A000041.

Cf. A001055, A005176, A027750, A056239, A072774, A100778, A295193, A306018, A318951, A319190, A319612, A322784, A322785, A322788, A322792.

\\ See A318951 for RowSumMats
row(n)={my(d=divisors(n)); vector(#d, i, RowSumMats(n/d[i], n, d[i]))}
{ for(n=1, 15, print(row(n))) } \\ Andrew Howroyd, Feb 02 2022

Terms a(28) and beyond from Andrew Howroyd, Feb 02 2022

Name edited by Peter Munn, Mar 05 2025

A322789 Irregular triangle read by rows where T(n,k) is the number of non-isomorphic uniform multiset partitions of a multiset with d = A027750(n,k) copies of each integer from 1 to n/d.

Original entry on oeis.org

1, 2, 2, 2, 2, 3, 4, 3, 2, 2, 4, 7, 6, 4, 2, 2, 4, 10, 8, 4, 3, 7, 3, 4, 12, 8, 4, 2, 2, 6, 32, 35, 31, 18, 6, 2, 2, 4, 21, 10, 4, 4, 47, 29, 4, 5, 49, 72, 19, 5, 2, 2, 6, 81, 170, 71, 24, 6, 2, 2, 6, 138, 478, 296, 32, 6, 4, 429, 76, 4, 4, 64, 14, 4

Offset: 1

Gus Wiseman, Dec 26 2018

A multiset partition is uniform if all parts have the same size.

			Triangle begins:
  1
  2  2
  2  2
  3  4  3
  2  2
  4  7  6  4
  2  2
  4 10  8  4
  3  7  3
  4 12  8  4
Non-isomorphic representatives of the multiset partitions counted under row 6:
{123456}           {112233}           {111222}           {111111}
{123}{456}         {112}{233}         {111}{222}         {111}{111}
{12}{34}{56}       {123}{123}         {112}{122}         {11}{11}{11}
{1}{2}{3}{4}{5}{6} {11}{22}{33}       {11}{12}{22}       {1}{1}{1}{1}{1}{1}
                   {11}{23}{23}       {12}{12}{12}
                   {12}{13}{23}       {1}{1}{1}{2}{2}{2}
                   {1}{1}{2}{2}{3}{3}

Andrew Howroyd, Table of n, a(n) for n = 1..338 (rows 1..75)

Row sums are A319056. First column is A000005.

Cf. A001055, A005176, A027750, A056239, A072774, A100778, A295193, A306017, A306018, A319190, A319612, A322784, A322785, A322787, A322792.

Terms a(28) and beyond from Andrew Howroyd, Feb 03 2022

Name edited by Peter Munn, Mar 05 2025

A326374 Irregular triangle read by rows where T(n,k) is the number of (d + 1)-uniform hypertrees spanning n + 1 vertices, where d = A027750(n,k).

Original entry on oeis.org

1, 3, 1, 16, 1, 125, 15, 1, 1296, 1, 16807, 735, 140, 1, 262144, 1, 4782969, 76545, 1890, 1, 100000000, 112000, 1, 2357947691, 13835745, 33264, 1, 61917364224, 1, 1792160394037, 3859590735, 270670400, 35135100, 720720, 1, 56693912375296, 1, 1946195068359375

Offset: 1

Gus Wiseman, Jul 03 2019

A hypertree is a connected hypergraph of density -1, where density is the sum of sizes of the edges minus the number of edges minus the number of vertices. A hypergraph is k-uniform if its edges all have size k. The span of a hypertree is the union of its edges.

			Triangle begins:
           1
           3          1
          16          1
         125         15          1
        1296          1
       16807        735        140          1
      262144          1
     4782969      76545       1890          1
   100000000     112000          1
  2357947691   13835745      33264          1
The T(4,2) = 15 hypertrees:
  {{1,4,5},{2,3,5}}
  {{1,4,5},{2,3,4}}
  {{1,3,5},{2,4,5}}
  {{1,3,5},{2,3,4}}
  {{1,3,4},{2,4,5}}
  {{1,3,4},{2,3,5}}
  {{1,2,5},{3,4,5}}
  {{1,2,5},{2,3,4}}
  {{1,2,5},{1,3,4}}
  {{1,2,4},{3,4,5}}
  {{1,2,4},{2,3,5}}
  {{1,2,4},{1,3,5}}
  {{1,2,3},{3,4,5}}
  {{1,2,3},{2,4,5}}
  {{1,2,3},{1,4,5}}

Alois P. Heinz, Rows n = 1..185, flattened

Row sums are A320444.
Column d = 1 is A000272.
Cf. A030019, A027750, A035053, A038041, A052888, A057625, A061095, A121860, A134954, A236696, A262843.

T:= n-> seq(n!/(d!*(n/d)!)*((n+1)/d)^(n/d-1), d=numtheory[divisors](n)):
seq(T(n), n=1..20);  # Alois P. Heinz, Aug 21 2019

Table[n!/(d!*(n/d)!)*((n+1)/d)^(n/d-1),{n,10},{d,Divisors[n]}]

T(n, k) = n!/(d! * (n/d)!) * ((n + 1)/d)^(n/d - 1), where d = A027750(n, k).

Edited by Peter Munn, Mar 05 2025

A329181 a(n) = n if n is 1 or prime; otherwise (1) let m = (concatenation of the two divisors in the middle of rows of A027750(n)), (2) if m is prime then a(n) = m, otherwise return to (1) with n=m.

Original entry on oeis.org

1, 2, 3, 211, 5, 23, 7, 223, 311, 773, 11, 21179, 13, 313, 1129, 3137, 17, 3449, 19, 59, 37, 211, 23, 223, 773, 3251, 313, 47, 29, 613, 31, 4373, 311, 21179, 1129, 3449, 37, 373, 313, 229, 41, 67, 43, 3137, 59, 223, 47, 4373, 131321, 33391, 317, 2333, 53

Offset: 1

David Cobac, Nov 07 2019

A term is a prime (or 1 for the first one) obtained by concatenating its two factors that are closest to its square root. Once they are concatenated (as strings), the process is iterated until concatenation gives a prime. Only composite numbers are processed.
At each iteration, we choose a couple (d, d') of divisors this way: n = d * d' and d = max({d >= 1 such that d|n and d<=sqrt(n)}), we replace n with the string concatenation of d and d' digits. The process ends with d = 1 (n is a prime).
This sequence is the balanced version of A316941.
Apparently it is only a conjecture that the process in the definition will alwats terminate. - N. J. A. Sloane, Feb 23 2020

			First three positive integers 1, 2, 3 do not change, so a(n)=n, for n <= 3.
4th term: sqrt(4)=2 and n=4=2*2 then n=22=2*11 and n=211 is a prime, so a(4)=211.
8th term: floor(sqrt(8))=2 and n=8=2*4 then n=24 but floor(sqrt(24))=4 so n=24=4*6 but floor(sqrt(46))=6 and 46's nearest factor to 6 is 2; thus 46=2*23 and 223 is a prime, so a(8)=223.
Thus a(8)=a(24)=a(46)=a(223)=223.

David Cobac, Table of n, a(n) for n = 1..590
David Cobac, C code

Same process as A316941 with different factors.

Cf. A002808 (the composite numbers), A027750 (the divisors of n).

C
```
/* See Cobac link. */
```

Array[If[! CompositeQ@ #, #, NestWhile[Block[{k = Floor@ Sqrt@ #}, While[Mod[#, k] != 0, k--]; FromDigits@ Flatten[IntegerDigits /@ {k, #/k}]] &, #, !PrimeQ@ # &]] &, 53] (* Michael De Vlieger, Nov 15 2019 *)

a(n) = {if (n==1, return (1)); if (isprime(n), return (n)); while (!isprime(n), my(d = divisors(n)); if (#d % 2 == 1, n = eval(concat(Str(d[#d\2+1]), Str(d[#d\2+1]))), n = eval(concat(Str(d[#d/2]), Str(d[#d/2+1]))));); n;} \\ Michel Marcus, Nov 15 2019

cp's OEIS Frontend

A378313 Numbers m such that such that Sum_{i=1..k} A027750(m,i)^i = m for some k <= A000005(m).

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A132588 Let b(k) be the k-th term of the flattened irregular array where the m-th row contains the positive divisors of m (b(k) = A027750(k)). Then a(n) = gcd(b(n), n).

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A292226 Composite numbers m (in increasing order) for which the m-th row polynomial of A027750 in rising powers is irreducible over the integers.

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A321725 Irregular triangle read by rows where T(n,k) is the number of d X d non-normal semi-magic squares with d = A027750(n,k) and sum of all entries equal to n.

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A322786 Irregular triangle read by rows where T(n,k) is the number of multiset partitions of a multiset with d = A027750(n,k) copies of each integer from 1 to n/d.

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A322787 Irregular triangle read by rows where T(n,k) is the number of non-isomorphic multiset partitions of a multiset with d = A027750(n, k) copies of each integer from 1 to n/d.

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PARI

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A322789 Irregular triangle read by rows where T(n,k) is the number of non-isomorphic uniform multiset partitions of a multiset with d = A027750(n,k) copies of each integer from 1 to n/d.

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