A027750 -id:A027750 - cp's OEIS frontend

A327482 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with mean d = A027750(n, k).

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 5, 4, 1, 1, 7, 1, 1, 7, 5, 1, 1, 1, 1, 11, 15, 12, 6, 1, 1, 1, 1, 15, 7, 1, 1, 30, 19, 1, 1, 22, 34, 8, 1, 1, 1, 1, 30, 58, 27, 9, 1, 1, 1, 1, 42, 84, 64, 10, 1, 1, 105, 37, 1, 1, 56, 11, 1

Offset: 1

Gus Wiseman, Sep 13 2019

			Triangle begins:
  1
  1  1
  1  1
  1  2  1
  1  1
  1  3  3  1
  1  1
  1  5  4  1
  1  7  1
  1  7  5  1
  1  1
  1 11 15 12  6  1
  1  1
  1 15  7  1
  1 30 19  1
  1 22 34  8  1

Row sums are A067538.
The version for subsets is A327481.
Cf. A027750, A066571, A237984, A240850, A327474, A327483, A327484.

Table[Length[Select[IntegerPartitions[n],Mean[#]==d&]],{n,20},{d,Divisors[n]}]

Name edited by Peter Munn, Mar 05 2025

A371783 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n that can be partitioned into d = A027750(n,k) blocks with equal sums.

Original entry on oeis.org

1, 2, 1, 3, 1, 5, 3, 1, 7, 1, 11, 6, 4, 1, 15, 1, 22, 14, 5, 1, 30, 10, 1, 42, 25, 6, 1, 56, 1, 77, 53, 30, 15, 7, 1, 101, 1, 135, 89, 8, 1, 176, 65, 21, 1, 231, 167, 55, 9, 1, 297, 1, 385, 278, 173, 28, 10, 1, 490, 1, 627, 480, 140, 91, 11, 1, 792, 343, 36, 1

Offset: 1

Gus Wiseman, Apr 14 2024

These could be called d-quanimous partitions, cf. A002219, A064914, A321452.

			Triangle begins:
    1
    2   1
    3   1
    5   3   1
    7   1
   11   6   4   1
   15   1
   22  14   5   1
   30  10   1
   42  25   6   1
   56   1
   77  53  30  15   7   1
  101   1
  135  89   8   1
  176  65  21   1
Row n = 6 counts the following partitions:
  (6)       (33)      (222)     (111111)
  (33)      (321)     (2211)
  (42)      (2211)    (21111)
  (51)      (3111)    (111111)
  (222)     (21111)
  (321)     (111111)
  (411)
  (2211)
  (3111)
  (21111)
  (111111)

Row lengths are A000005.
Column k = 1 is A000041.
Inserting zeros gives A371954.
Row sums are A372121.
A002219 (aerated) counts biquanimous partitions, ranks A357976.
A237258 aerated counts biquanimous strict partitions, ranks A357854.
A321142 and A371794 count non-biquanimous strict partitions.
A321451 counts non-quanimous partitions, ranks A321453.
A321452 counts quanimous partitions, ranks A321454.
A371736 counts non-quanimous strict partitons, complement A371737.
A371781 lists numbers with biquanimous prime signature, complement A371782.
A371789 counts non-quanimous sets, differences A371790.
A371796 counts quanimous sets, differences A371797.
Cf. A006827, A027750, A035470, A064914, A321455, A365543, A371791, A371795.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
facs[n_]:=If[n<=1,{{}}, Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]], {d,Rest[Divisors[n]]}]];
Table[Length[Select[IntegerPartitions[n], Select[facs[Times@@Prime/@#], Length[#]==k&&SameQ@@hwt/@#&]!={}&]],{n,1,8},{k,Divisors[n]}]

More terms from Jinyuan Wang, Feb 13 2025

Name edited by Peter Munn, Mar 05 2025

A240698 Partial sums of divisors of n, cf. A027750.

Original entry on oeis.org

1, 1, 3, 1, 4, 1, 3, 7, 1, 6, 1, 3, 6, 12, 1, 8, 1, 3, 7, 15, 1, 4, 13, 1, 3, 8, 18, 1, 12, 1, 3, 6, 10, 16, 28, 1, 14, 1, 3, 10, 24, 1, 4, 9, 24, 1, 3, 7, 15, 31, 1, 18, 1, 3, 6, 12, 21, 39, 1, 20, 1, 3, 7, 12, 22, 42, 1, 4, 11, 32, 1, 3, 14, 36, 1, 24, 1

Offset: 1

Reinhard Zumkeller, Apr 10 2014

Triangle read by rows in which row n lists the partial sums of divisors of n. - Omar E. Pol, Apr 12 2014

			.    n |  n-th row of A240698   |  n-th row of A027750
.  ----+------------------------+---------------------
.    1 |  1                     |  1
.    2 |  1, 3                  |  1, 2
.    3 |  1, 4                  |  1, 3
.    4 |  1, 3, 7               |  1, 2, 4
.    5 |  1, 6                  |  1, 5
.    6 |  1, 3, 6, 12           |  1, 2, 3, 6
.    7 |  1, 8                  |  1, 7
.    8 |  1, 3, 7, 15           |  1, 2, 4, 8
.    9 |  1, 4, 13              |  1, 3, 9
.   10 |  1, 3, 8, 18           |  1, 2, 5, 10
.   11 |  1, 12                 |  1, 11
.   12 |  1, 3, 6, 10, 16, 28   |  1, 2, 3, 4, 6, 12
.   13 |  1, 14                 |  1, 13 .

Reinhard Zumkeller, Rows n = 1..1000 of table, flattened

Cf. A000005 (row lengths), A240694.

Cf. A000203, A001065, A027750, A064510, A194472.

a240698 n k = a240698_tabf !! (n-1) !! (k-1)
a240698_row n = a240698_tabf !! (n-1)
a240698_tabf = map (scanl1 (+)) a027750_tabf

Table[Accumulate[Divisors[n]],{n,30}]//Flatten (* Harvey P. Dale, Dec 30 2019 *)

row(n) = my(d=divisors(n)); vector(#d, k, sum(i=1, k, d[i])); \\ Michel Marcus, Jan 24 2022

T(n,1) = 1, T(n,k) = T(n,k-1) + A027750(n,k), 1 < k <= n.
T(n,1) = 1;
T(n,A000005(n)) = A000203(n);
T(n,A000005(n)-1) = A001065(n), n > 1.

A240694 Partial products of divisors of n, cf. A027750.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 8, 1, 5, 1, 2, 6, 36, 1, 7, 1, 2, 8, 64, 1, 3, 27, 1, 2, 10, 100, 1, 11, 1, 2, 6, 24, 144, 1728, 1, 13, 1, 2, 14, 196, 1, 3, 15, 225, 1, 2, 8, 64, 1024, 1, 17, 1, 2, 6, 36, 324, 5832, 1, 19, 1, 2, 8, 40, 400, 8000, 1, 3, 21, 441, 1, 2

Offset: 1

Reinhard Zumkeller, Apr 10 2014

Triangle read by rows in which row n lists the partial products of divisors of n. - Omar E. Pol, Apr 12 2014

			.    n |  n-th row of A240694     |  n-th row of A027750
.  ----+--------------------------+---------------------
.    1 |  1                       |  1
.    2 |  1, 2                    |  1, 2
.    3 |  1, 3                    |  1, 3
.    4 |  1, 2, 8                 |  1, 2, 4
.    5 |  1, 5                    |  1, 5
.    6 |  1, 2, 6, 36             |  1, 2, 3, 6
.    7 |  1, 7                    |  1, 7
.    8 |  1, 2, 8, 64             |  1, 2, 4, 8
.    9 |  1, 3, 27                |  1, 3, 9
.   10 |  1, 2, 10, 100           |  1, 2, 5, 10
.   11 |  1, 11                   |  1, 11
.   12 |  1, 2, 6, 24, 144, 1728  |  1, 2, 3, 4, 6, 12
.   13 |  1, 13                   |  1, 13 .

Reinhard Zumkeller, Rows n = 1..1000 of table, flattened

Cf. A000005 (row lengths), A007955, A020639, A027750, A240698.

a240694 n k = a240694_tabf !! (n-1) !! (k-1)
a240694_row n = a240694_tabf !! (n-1)
a240694_tabf = map (scanl1 (*)) a027750_tabf

Table[FoldList[Times,Divisors[n]],{n,30}]//Flatten (* Harvey P. Dale, Jul 29 2021 *)

row(n) = my(d=divisors(n)); vector(#d, k, prod(i=1, k, d[i])); \\ Michel Marcus, Jan 24 2022

T(n,1) = 1, T(n,k) = T(n,k-1) * A027750(n,k), 1 < k <= n.
T(n,1) = 1;
T(n,2) = A020639(n), n > 1;
T(n,A000005(n)) = A007955(n);
T(n,A000005(n)-1) = A007956(n), n > 1.

A317748 Irregular triangle where T(n,k) is the number of factorizations of n into factors > 1 with GCD d = A027750(n, k).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 2, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 3, 1, 0, 1, 0, 1, 2, 0, 1, 0, 0, 1, 0, 1, 2, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 3, 3, 0, 0, 0, 0, 0, 1, 0, 1, 1

Offset: 1

Gus Wiseman, Aug 06 2018

			Triangle begins:
   1:  0
   2:  0  1
   3:  0  1
   4:  0  1  1
   5:  0  1
   6:  1  0  0  1
   7:  0  1
   8:  0  2  0  1
   9:  0  1  1
  10:  1  0  0  1
  11:  0  1
  12:  2  1  0  0  0  1
  13:  0  1
  14:  1  0  0  1
  15:  1  0  0  1
  16:  0  3  1  0  1
  17:  0  1
  18:  2  0  1  0  0  1
  19:  0  1
  20:  2  1  0  0  0  1

Row lengths are A000005. Row sums are A001055. First column is A281116. Number of nonzero terms in each row is A317751.
Cf. A007562, A007716, A014963, A027750, A045778, A050370, A162247, A289509, A303386, A303546.
Cf. A317752, A317755, A317757.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
goc[n_,m_]:=Length[Select[facs[n],And[And@@(Divisible[#,m]&/@#),GCD@@(#/m)==1]&]];
Table[goc[n,d],{n,30},{d,Divisors[n]}]

Name edited by Peter Munn, Mar 05 2025

A322792 Irregular triangle read by rows where T(n,k) = A002110(n/d)^d, where d = A027750(n,k) and A002110(m) is the product of the first m primes.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 26 2018

A reordering of A100778 (powers of primorials), these are the Heinz numbers of uniform integer partitions of length n 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).

			Triangle begins:
           2
           6          4
          30          8
         210         36         16
        2310         32
       30030        900        216         64
      510510        128
     9699690      44100       1296        256
   223092870      27000        512
  6469693230    5336100       7776       1024
Corresponding triangle of integer partitions whose Heinz numbers belong to the triangle begins:
  (1)
  (21)        (11)
  (321)       (111)
  (4321)      (2211)      (1111)
  (54321)     (11111)
  (654321)    (332211)    (222111)    (111111)
  (7654321)   (1111111)
  (87654321)  (44332211)  (22221111)  (11111111)
  (987654321) (333222111) (111111111)

First column is A002110.

Cf. A000961, A001597, A002110, A007947, A027750, A025487, A047966, A055932, A056239, A072774, A100778, A304250, A322793.

Table[Product[Prime[i]^d,{i,n/d}],{n,12},{d,Divisors[n]}]

Name edited by Peter Munn, Mar 05 2025

A321724 Irregular triangle read by rows where T(n,k) is the number of non-isomorphic non-normal semi-magic square multiset partitions of weight n and length d = A027750(n, k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 3, 5, 1, 1, 5, 1, 1, 3, 7, 1, 1, 1, 1, 4, 9, 12, 11, 1, 1, 1, 1, 4, 15, 1, 1, 13, 31, 1, 1, 5, 43, 22, 1, 1, 1, 1, 5, 22, 103, 30, 1, 1, 1, 1, 6, 106, 264, 42, 1, 1, 30, 383, 1, 1, 6, 56, 1, 1, 1, 1, 7, 45, 321, 2804, 1731, 77, 1

Offset: 1

Gus Wiseman, Nov 18 2018

Also the number of nonnegative integer square matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, with row sums and column sums all equal to d.
A non-normal semi-magic square multiset partition of weight n is a multiset partition of weight n whose part sizes and vertex degrees are all equal to d, for some d|n.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Triangle begins:
  1
  1 1
  1 1
  1 2 1
  1 1
  1 2 3 1
  1 1
  1 3 5 1
  1 5 1
  1 3 7 1
Inequivalent representatives of the T(10,3) = 7 semi-magic squares (zeros not shown):
  [2    ] [2    ] [2    ] [2    ] [2    ] [11   ] [11   ]
  [ 2   ] [ 2   ] [ 2   ] [ 11  ] [ 11  ] [11   ] [1 1  ]
  [  2  ] [  2  ] [  11 ] [ 11  ] [ 1 1 ] [  11 ] [ 1 1 ]
  [   2 ] [   11] [  1 1] [   11] [  1 1] [  1 1] [  1 1]
  [    2] [   11] [   11] [   11] [   11] [   11] [   11]

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

Row sums are A321721.
Cf. A006052, A007016, A007716, A027750, A057150, A120732, A271103, A319056, A319616.
Cf. A321718, A321719, A321722, A333733.

T(n,k) = A333733(d, n/d), where d = A027750(n, k). - Andrew Howroyd, Apr 11 2020

a(28)-a(39) from Chai Wah Wu, Jan 16 2019
Terms a(40) and beyond from Andrew Howroyd, Apr 11 2020
Edited by Peter Munn, Mar 05 2025

A056534 Mapping from the ordering by product (A027750, A056538) to the ordering by sum (A002260, A004736) of ordered pairs (a,b), a>=1, b>=1.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 5, 10, 11, 15, 16, 8, 9, 21, 22, 28, 29, 12, 14, 36, 37, 13, 45, 46, 17, 20, 55, 56, 66, 67, 23, 18, 19, 27, 78, 79, 91, 92, 30, 35, 105, 106, 24, 26, 120, 121, 38, 25, 44, 136, 137, 153, 154, 47, 31, 34, 54, 171, 172, 190, 191, 57, 32, 33, 65, 210, 211, 39

Offset: 1

Antti Karttunen, Jun 20 2000

nonn

			The "ordering by sum": (1,1),(1,2),(2,1),(1,3),(2,2),(3,1),(1,4),(2,3),(3,2),(4,1),...
The "ordering by product": (1,1),(1,2),(2,1),(1,3),(3,1),(1,4),(2,2),(4,1),(1,5),(5,1),...

Inverse: A056535.

ordered_pair_perm := proc(upto_n) local a,i,j; a := []; for i from 1 to upto_n do for j in sort(divisors(i)) do a := [op(a),binomial(((i/j) + j - 1),2)+j]; od; od; RETURN(a); end;

max = 21; A056534 = {}; For[i = 1, i <= max, i++, Do[ AppendTo[ A056534, Binomial[i/j + j - 1, 2] + j], {j, Divisors[i]}]]; A056534 (* Jean-François Alcover, Oct 05 2012, after Maple *)

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

A331010 Lexicographically earliest sequence of distinct positive terms that can be viewed as an irregular table where the n-th row has A000005(a(n)) terms and T(n, k) is a multiple of the k-th divisor of a(n) (=A027750(a(n), k)).

Original entry on oeis.org

1, 2, 4, 3, 6, 8, 5, 9, 7, 10, 12, 18, 11, 14, 16, 24, 13, 15, 17, 21, 27, 19, 28, 20, 22, 25, 30, 23, 26, 33, 32, 36, 48, 29, 34, 39, 42, 45, 54, 31, 44, 35, 38, 49, 56, 37, 40, 52, 64, 80, 41, 46, 51, 60, 66, 72, 84, 96, 43, 65, 47, 57, 50, 75, 53, 68, 55

Offset: 1

Rémy Sigrist, Jan 06 2020

This sequence is a permutation of the natural numbers as each row starts with the least value not yet in the sequence.

The prime numbers appear in ascending order.

			The first terms and rows are:
  n   a(n)  row(n)
  --  ----  -------------------------
   1     1  [1]
   2     2  [2, 4]
   3     4  [3, 6, 8]
   4     3  [5, 9]
   5     6  [7, 10, 12, 18]
   6     8  [11, 14, 16, 24]
   7     5  [13, 15]
   8     9  [17, 21, 27]
   9     7  [19, 28]
  10    10  [20, 22, 25, 30]
  11    12  [23, 26, 33, 32, 36, 48]
  12    18  [29, 34, 39, 42, 45, 54]
  13    11  [31, 44]
  14    14  [35, 38, 49, 56]
  15    16  [37, 40, 52, 64, 80]
  16    24  [41, 46, 51, 60, 66, 72, 84, 96]

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers
Rémy Sigrist, PARI program for A331010

See A331016, A331018 and A331026 for similar sequences.

Cf. A000005, A027750.

PARI
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See Links section.
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cp's OEIS Frontend

A327482 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with mean d = A027750(n, k).

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A371783 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n that can be partitioned into d = A027750(n,k) blocks with equal sums.

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A240698 Partial sums of divisors of n, cf. A027750.

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A240694 Partial products of divisors of n, cf. A027750.

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A317748 Irregular triangle where T(n,k) is the number of factorizations of n into factors > 1 with GCD d = A027750(n, k).

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A322792 Irregular triangle read by rows where T(n,k) = A002110(n/d)^d, where d = A027750(n,k) and A002110(m) is the product of the first m primes.

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A321724 Irregular triangle read by rows where T(n,k) is the number of non-isomorphic non-normal semi-magic square multiset partitions of weight n and length d = A027750(n, k).

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A056534 Mapping from the ordering by product (A027750, A056538) to the ordering by sum (A002260, A004736) of ordered pairs (a,b), a>=1, b>=1.

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