cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A323306 Heinz numbers of integer partitions that can be arranged into a matrix with equal row-sums and equal column-sums.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 100, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193
Offset: 1

Views

Author

Gus Wiseman, Jan 13 2019

Keywords

Comments

First differs from A137944 in lacking 120.
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Examples

			6480 belongs to the sequence because it is the Heinz number of (3,2,2,2,2,1,1,1,1), which can be arranged in the following ways:
  [1 1 3] [1 2 2] [1 2 2] [1 3 1] [2 1 2] [2 1 2] [2 2 1] [2 2 1] [3 1 1]
  [2 2 1] [1 2 2] [3 1 1] [2 1 2] [1 3 1] [2 1 2] [1 1 3] [2 2 1] [1 2 2]
  [2 2 1] [3 1 1] [1 2 2] [2 1 2] [2 1 2] [1 3 1] [2 2 1] [1 1 3] [1 2 2]

Links

Crossrefs

Complement of A323304.
Cf. A006052, A007016, A056239, A112798, A120733, A319056, A319066, A321719.
Cf. A323300, A323302, A323347, A323348, A323349.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
Select[Range[100],!Select[ptnmats[#],And[SameQ@@Total/@#,SameQ@@Total/@Transpose[#]]&]=={}&]

A323349 Number of positive integer matrices with entries summing to n, with equal row-sums and equal column-sums.

Original entry on oeis.org

1, 1, 3, 3, 6, 3, 11, 3, 12, 6, 13, 3, 52, 3, 15, 30, 57, 3, 156, 3, 238, 129, 19, 3, 2221, 6, 21, 415, 3114, 3, 14921, 3, 12853, 1044, 25, 6219, 164743, 3, 27, 2220, 851476, 3, 954088, 3, 434106, 3326714, 31, 3, 24648724, 6, 22309800, 7269, 2737618, 3, 69823653
Offset: 0

Views

Author

Gus Wiseman, Jan 13 2019

Keywords

Comments

Also the number of non-normal semi-magic rectangles summing to n with no zeros.
Matrices must be of size m X k where m, k are divisors of n and mk <= n. This implies that a(p) = 3 for p prime, since the only allowable matrices must be of size 1 X 1, 1 X p or p X 1 with only one way to fill in the entries for each matrix size. Similarly, a(p^2) = 6 with additional allowable matrices of sizes 1 X p^2, p^2 X 1 and p X p, again with only one way to fill in the entries for each size. - Chai Wah Wu, Jan 13 2019

Examples

			The a(6) = 11 matrices:
  [6] [3 3] [2 2 2] [1 1 1 1 1 1]
.
  [3] [1 2] [2 1] [1 1 1]
  [3] [2 1] [1 2] [1 1 1]
.
  [2] [1 1]
  [2] [1 1]
  [2] [1 1]
.
  [1]
  [1]
  [1]
  [1]
  [1]
  [1]

Links

Crossrefs

Cf. A000219, A006052, A120733, A305551, A319056, A321719, A323295, A323300, A323302, A323306, A323347, A323349.

Programs

  • Mathematica
    Table[Length[Select[Join@@Table[Partition[cmp,d],{cmp,Join@@Permutations/@IntegerPartitions[n]},{d,Divisors[Length[cmp]]}],And[SameQ@@Total/@#,SameQ@@Total/@Transpose[#]]&]],{n,10}]

Formula

a(p) = 3 and a(p^2) = 6 for p prime (see comment). - Chai Wah Wu, Jan 13 2019

Extensions

a(21)-a(31) from Chai Wah Wu, Jan 13 2019
a(32)-a(53) from Chai Wah Wu, Jan 14 2019
a(54) from Chai Wah Wu, Jan 16 2019

A323302 Number of ways to arrange the parts of the integer partition with Heinz number n into a matrix with equal row-sums and equal column-sums.

Original entry on oeis.org

1, 1, 1, 2, 1, 0, 1, 2, 2, 0, 1, 0, 1, 0, 0, 3, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 2, 0, 0, 0, 2, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 4, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 3, 0, 1, 0, 0, 0, 0
Offset: 1

Views

Author

Gus Wiseman, Jan 13 2019

Keywords

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Examples

			The a(900) = 12 matrix-arrangements of (3,3,2,2,1,1):
  [1 2 3] [1 3 2] [2 1 3] [2 3 1] [3 1 2] [3 2 1]
  [3 2 1] [3 1 2] [2 3 1] [2 1 3] [1 3 2] [1 2 3]
.
  [1 3] [1 3] [2 2] [2 2] [3 1] [3 1]
  [2 2] [3 1] [1 3] [3 1] [1 3] [2 2]
  [3 1] [2 2] [3 1] [1 3] [2 2] [1 3]

Crossrefs

Positions of zeros are a superset of A106543.
Cf. A000005, A001222, A006052, A007016, A008480, A056239, A112798, A120733, A319056, A321719, A321721.
Cf. A323300, A323303, A323305, A323306, A323347, A323349.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
Table[Length[Select[ptnmats[n],And[SameQ@@Total/@#,SameQ@@Total/@Transpose[#]]&]],{n,100}]

A323304 Heinz numbers of integer partitions that cannot be arranged into a matrix with equal row-sums and equal column-sums.

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 102, 104, 105
Offset: 1

Views

Author

Gus Wiseman, Jan 13 2019

Keywords

Comments

The first term of this sequence absent from A106543 is 144.
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Links

Crossrefs

Complement of A323306.
Cf. A006052, A007016, A008480, A056239, A112798, A120733, A319056, A321719, A321721.
Cf. A323300, A323302, A323305, A323347, A323348, A323349.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
Select[Range[2,1000],Select[ptnmats[#],And[SameQ@@Total/@#,SameQ@@Total/@Transpose[#]]&]=={}&]

A323348 Number of integer partitions of n whose parts cannot be arranged into a (not necessarily square) matrix with equal row-sums and equal column-sums.

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 6, 13, 17, 27, 36, 54, 66, 99, 128, 169, 221, 295, 367, 488, 610, 779, 993, 1253, 1525, 1955, 2426, 2986, 3684, 4563, 5519, 6840, 8298, 10097, 12298, 14874, 17716, 21635, 26002, 31105, 37081, 44581, 52916, 63259, 74852, 88703, 105543, 124752, 145740, 173522, 203999, 239737, 280424, 329929
Offset: 0

Views

Author

Gus Wiseman, Jan 13 2019

Keywords

Examples

			The a(8) = 17 integer partitions:
  (53), (62), (71),
  (332), (422), (431), (521), (611),
  (3221), (4211), (5111),
  (22211), (32111), (41111),
  (221111), (311111),
  (2111111).

Links

Crossrefs

A000041(n) = A323347(n) + a(n).
Cf. A006052, A007016, A120733, A319056, A319066, A321719.
Cf. A323300, A323302, A323304, A323306, A323349.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
Table[Length[Select[IntegerPartitions[n],Select[ptnmats[Times@@Prime/@#],And[SameQ@@Total/@#,SameQ@@Total/@Transpose[#]]&]=={}&]],{n,10}]

Extensions

a(17)-a(53) from Chai Wah Wu, Jan 15 2019

A323303 Number of ways to arrange the prime indices of n into a matrix with equal column-sums.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 2, 3, 1, 6, 1, 2, 2, 2, 2, 10, 1, 2, 2, 4, 1, 6, 1, 3, 3, 2, 1, 5, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 12, 1, 2, 3, 4, 2, 6, 1, 3, 2, 6, 1, 10, 1, 2, 3, 3, 2, 6, 1, 5, 3, 2, 1, 12, 2, 2
Offset: 1

Views

Author

Gus Wiseman, Jan 17 2019

Keywords

Comments

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.

Examples

			The a(90) = 16 matrix-arrangements of (3,2,2,1) with equal column-sums:
  [1 2] [2 1] [2 3] [3 2]
  [3 2] [2 3] [2 1] [1 2]
.
  [1] [1] [1] [2] [2] [2] [2] [2] [2] [3] [3] [3]
  [2] [2] [3] [1] [1] [2] [2] [3] [3] [1] [2] [2]
  [2] [3] [2] [2] [3] [1] [3] [1] [2] [2] [1] [2]
  [3] [2] [2] [3] [2] [3] [1] [2] [1] [2] [2] [1]

Crossrefs

Cf. A006052, A008480, A056239, A112798, A118914, A120733, A319056, A321719.
Cf. A323295, A323300, A323302, A323305, A323306, A323347, A323349.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
Table[Length[Select[ptnmats[n],SameQ@@Total/@Transpose[#]&]],{n,100}]

A323524 Number of integer partitions of n whose parts can be arranged into a square matrix with equal row and column sums.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 3, 1, 5, 1, 4, 4, 6, 1, 10, 1, 7, 10, 6, 1, 24, 2, 7, 22, 18, 1, 38, 1, 35, 43, 9, 6, 124, 1, 10, 77, 158, 1, 110, 1, 285, 186, 12, 1, 742, 2, 170, 203, 1110, 1, 285, 480, 2115, 306, 15, 1
Offset: 0

Views

Author

Gus Wiseman, Jan 17 2019

Keywords

Examples

			The a(12) = 5 integer partitions are (12), (5,5,1,1), (4,4,2,2), (3,3,3,3), (2,2,2,1,1,1,1,1,1). For example, such a matrix for (2,2,2,1,1,1,1,1,1) is:
  [1 1 2]
  [2 1 1]
  [1 2 1]

Crossrefs

Cf. A006052, A007016, A103198, A120732, A321719, A321722, A321725, A323306, A323347, A323349, A323523.

Formula

a(p) = 1 and a(p^2) = 2 for p prime (see comment in A323349). - Chai Wah Wu, Jan 20 2019

Extensions

a(16)-a(59) from Chai Wah Wu, Jan 20 2019
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.