A322792 -id:A322792 - cp's OEIS frontend

A100778 Integer powers of primorial numbers.

1, 2, 4, 6, 8, 16, 30, 32, 36, 64, 128, 210, 216, 256, 512, 900, 1024, 1296, 2048, 2310, 4096, 7776, 8192, 16384, 27000, 30030, 32768, 44100, 46656, 65536, 131072, 262144, 279936, 510510, 524288, 810000, 1048576, 1679616, 2097152, 4194304, 5336100

Offset: 1

Amarnath Murthy, Nov 28 2004

Smallest squarefree numbers or their powers with distinct prime signatures. Or least numbers with prime signatures (p*q*r*...)^k, where p,q,r,... are primes and k is a whole number.
Also Heinz numbers of uniform integer partitions whose union is an initial interval of positive integers. An integer partition is uniform if all parts appear with the same multiplicity. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k). The sequence of all uniform integer partitions whose Heinz numbers belong to the sequence begins: (1), (11), (12), (111), (1111), (123), (11111), (1122), (111111), (1111111), (1234), (111222), (11111111), (111111111), (112233), (1111111111). - Gus Wiseman, Dec 26 2018
From Amiram Eldar, Sep 26 2023: (Start)
Intersection of A025487 and A072774.
The distinct terms of A046523(A072774(n)) in ascending orders.
The k-th power of the n-th primorial number, A002110(n)^k, has (k+1)^n divisors which are the set of the (k+1)-free prime(n)-smooth numbers. (End)

			10 is not a term as 6 is a member with the same prime signature 10 > 6.
216 is a term as 216 = (2*3)^3. 243 is not a term as 32 represents that prime signature.

David A. Corneth, Table of n, a(n) for n = 1..8606 (terms <= 10^1000)

Cf. A000961, A001597, A002110, A007947, A025487, A046523, A055932, A056239, A057588, A072774, A072777, A112798, A304250, A319169, A322792, A322793.

unintQ[n_]:=And[SameQ@@Last/@FactorInteger[n],Length[FactorInteger[n]]==PrimePi[FactorInteger[n][[-1,1]]]];
Select[Range[1000],unintQ] (* Gus Wiseman, Dec 26 2018 *)

Sum_{n>=1} 1/a(n) = 1 + Sum_{n>=1} 1/A057588(n) = 2.2397359032... - Amiram Eldar, Oct 20 2020; corrected by Hal M. Switkay and Amiram Eldar, Apr 12 2021

More terms and simpler definition from Ray Chandler, Nov 29 2004

A322784 Number of multiset partitions of uniform multisets of size n whose union is an initial interval of positive integers.

Original entry on oeis.org

1, 1, 4, 8, 29, 59, 311, 892, 4983, 21863, 126813, 678626, 4446565, 27644538, 195561593, 1384705697, 10613378402, 82864870101, 686673571479, 5832742205547, 51897707277698, 474889512098459, 4514467567213008, 44152005855085601, 446355422070799305, 4638590359349994120

Offset: 0

Gus Wiseman, Dec 26 2018

nonn

A multiset is uniform if all multiplicities are equal.
Also the number of factorizations into factors > 1 of primorial powers k in A100778 with sum of prime indices A056239(k) equal to n.
a(n) is the number of nonequivalent nonnegative integer matrices without zero rows or columns with equal column sums and total sum n up to permutation of rows. - Andrew Howroyd, Jan 11 2020

			The a(1) = 1 through a(4) = 29 multiset partitions:
  {{1}}   {{1,1}}     {{1,1,1}}       {{1,1,1,1}}
          {{1,2}}     {{1,2,3}}       {{1,1,2,2}}
         {{1},{1}}   {{1},{1,1}}      {{1,2,3,4}}
         {{1},{2}}   {{1},{2,3}}     {{1},{1,1,1}}
                     {{2},{1,3}}     {{1,1},{1,1}}
                     {{3},{1,2}}     {{1},{1,2,2}}
                    {{1},{1},{1}}    {{1,1},{2,2}}
                    {{1},{2},{3}}    {{1,2},{1,2}}
                                     {{1},{2,3,4}}
                                     {{1,2},{3,4}}
                                     {{1,3},{2,4}}
                                     {{1,4},{2,3}}
                                     {{2},{1,1,2}}
                                     {{2},{1,3,4}}
                                     {{3},{1,2,4}}
                                     {{4},{1,2,3}}
                                    {{1},{1},{1,1}}
                                    {{1},{1},{2,2}}
                                    {{1},{2},{1,2}}
                                    {{1},{2},{3,4}}
                                    {{1},{3},{2,4}}
                                    {{1},{4},{2,3}}
                                    {{2},{2},{1,1}}
                                    {{2},{3},{1,4}}
                                    {{2},{4},{1,3}}
                                    {{3},{4},{1,2}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{2},{2}}
                                   {{1},{2},{3},{4}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Row sums of A322786.

Cf. A001055, A005176, A056239, A072774, A100778, A219727, A295193, A306017 (unlabeled version), A319190, A319612, A322785, A322786, A322792.

u[n_,k_]:=u[n,k]=If[n==1,1,Sum[u[n/d,d],{d,Select[Rest[Divisors[n]],#<=k&]}]];
Table[Sum[u[Array[Prime,d,1,Times]^(n/d),Array[Prime,d,1,Times]^(n/d)],{d,Divisors[n]}],{n,12}]

a(n) = Sum_{d|n} A001055(A002110(n/d)^d).

a(n) = Sum_{d|n} A219727(n/d, d). - Andrew Howroyd, Jan 11 2020

a(14)-a(15) from Alois P. Heinz, Jan 16 2019

Terms a(16) and beyond from Andrew Howroyd, Jan 11 2020

A322788 Irregular triangle read by rows where T(n,k) is the number of 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, 5, 4, 3, 2, 2, 27, 11, 6, 4, 2, 2, 142, 29, 8, 4, 282, 12, 3, 1073, 101, 8, 4, 2, 2, 32034, 1581, 234, 75, 20, 6, 2, 2, 136853, 2660, 10, 4, 1527528, 1985, 91, 4, 4661087, 64596, 648, 20, 5, 2, 2, 227932993, 1280333, 41945, 231, 28, 6

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
     5    4    3
     2    2
    27   11    6    4
     2    2
   142   29    8    4
   282   12    3
  1073  101    8    4
The multiset partitions counted under row 6:
  {123456}          {112233}          {111222}          {111111}
  {123}{456}        {112}{233}        {111}{222}        {111}{111}
  {124}{356}        {113}{223}        {112}{122}        {11}{11}{11}
  {125}{346}        {122}{133}        {11}{12}{22}      {1}{1}{1}{1}{1}{1}
  {126}{345}        {123}{123}        {12}{12}{12}
  {134}{256}        {11}{22}{33}      {1}{1}{1}{2}{2}{2}
  {135}{246}        {11}{23}{23}
  {136}{245}        {12}{12}{33}
  {145}{236}        {12}{13}{23}
  {146}{235}        {13}{13}{22}
  {156}{234}        {1}{1}{2}{2}{3}{3}
  {12}{34}{56}
  {12}{35}{46}
  {12}{36}{45}
  {13}{24}{56}
  {13}{25}{46}
  {13}{26}{45}
  {14}{23}{56}
  {14}{25}{36}
  {14}{26}{35}
  {15}{23}{46}
  {15}{24}{36}
  {15}{26}{34}
  {16}{23}{45}
  {16}{24}{35}
  {16}{25}{34}
  {1}{2}{3}{4}{5}{6}

Andrew Howroyd, Table of n, a(n) for n = 1..482 (rows 1..100)

Row sums are A322785. First column is A038041.

Cf. A001055, A005176, A027750, A056239, A072774, A100778, A295193, A306017, A319190, A319612, A322784, A322785, A322786, A322789, A322792.

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[mps[Join@@Table[Range[n/d],{d}]],SameQ@@Length/@#&]],{n,10},{d,Divisors[n]}]

T(n,k) = A322794(A002110(n/d)^d), where d = A027750(n,k).

More terms from Alois P. Heinz, Jan 30 2019
Terms a(38) and beyond from Andrew Howroyd, Feb 03 2022
Edited by Peter Munn, Mar 05 2025

A322793 Proper powers of primorial numbers.

Original entry on oeis.org

4, 8, 16, 32, 36, 64, 128, 216, 256, 512, 900, 1024, 1296, 2048, 4096, 7776, 8192, 16384, 27000, 32768, 44100, 46656, 65536, 131072, 262144, 279936, 524288, 810000, 1048576, 1679616, 2097152, 4194304, 5336100, 8388608, 9261000

Offset: 1

Gus Wiseman, Dec 26 2018

nonn

A primorial number is a product of the first n primes, for some n.

Also Heinz numbers of non-strict uniform integer partitions whose union is an initial interval of positive integers. An integer partition is uniform if all parts appear with the same multiplicity. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of all non-strict uniform integer partitions whose Heinz numbers belong to the sequence begins: (11), (111), (1111), (11111), (2211), (111111), (1111111), (222111), (11111111), (111111111), (332211), (1111111111), (22221111).

Michael De Vlieger, Table of n, a(n) for n = 1..8255

Cf. A000961, A001597, A001694, A002110, A025487, A047966, A055932, A056239, A072774, A072777, A100778, A112798, A304250, A322792.

unintpropQ[n_]:=And[SameQ@@Last/@FactorInteger[n],FactorInteger[n][[1,2]]>1,Length[FactorInteger[n]]==PrimePi[FactorInteger[n][[-1,1]]]];
Select[Range[10000],unintpropQ]
(* Second program: *)
nn = 2^24; k = 1; P = 2; Union@ Reap[While[j = 2; While[P^j < nn, Sow[P^j]; j++]; j > 2, k++; P *= Prime[k]]][[-1, 1]] (* Michael De Vlieger, Oct 04 2023 *)

Sum_{n>=1} 1/a(n) = Sum_{k>=1} 1/(A002110(k)*(A002110(k)-1)) = 0.53450573145072369022... . - Amiram Eldar, Mar 10 2024

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

A384180 Irregular triangle read by rows where row n lists the Heinz numbers of all uniform (equal multiplicities) and normal (covering an initial interval) multisets of length n.

Original entry on oeis.org

2, 4, 6, 8, 30, 16, 36, 210, 32, 2310, 64, 216, 900, 30030, 128, 510510, 256, 1296, 44100, 9699690, 512, 27000, 223092870, 1024, 7776, 5336100, 6469693230, 2048, 200560490130, 4096, 46656, 810000, 9261000, 901800900, 7420738134810, 8192, 304250263527210

Offset: 1

Gus Wiseman, May 25 2025

A permutation of A100778 (powers of primorials).
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
An integer partition is uniform iff all parts appear with the same multiplicity, and normal iff it covers an initial interval of positive integers.

			The uniform normal multisets of length 6 are: {1,1,1,1,1,1}, {1,1,1,2,2,2}, {1,1,2,2,3,3}, {1,2,3,4,5,6}, so row 6 is: 64, 216, 900, 30030.
Triangle begins:
    2
    4       6
    8      30
   16      36    210
   32    2310
   64     216    900    30030
  128  510510
  256    1296  44100  9699690

Row lengths are A000005.
Final term in each row is A002110.
The union is A100778.
Reversing rows gives A322792.
For just normal multisets we have A324939.
A047966 counts uniform partitions.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A098859 counts Wilf partitions, ranks A130091, conjugate A383512.
A381431 is the section-sum transform.
Cf. A000009, A001694, A056239, A106529, A112798, A116540, A122111, A258280, A325326, A325337.

Table[Table[Times@@Prime/@Range[d]^(n/d),{d,Divisors[n]}],{n,10}]

cp's OEIS Frontend

A100778 Integer powers of primorial numbers.

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A322784 Number of multiset partitions of uniform multisets of size n whose union is an initial interval of positive integers.

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A322788 Irregular triangle read by rows where T(n,k) is the number of uniform multiset partitions of a multiset with d = A027750(n,k) copies of each integer from 1 to n/d.

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A322793 Proper powers of primorial numbers.

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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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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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A384180 Irregular triangle read by rows where row n lists the Heinz numbers of all uniform (equal multiplicities) and normal (covering an initial interval) multisets of length n.