A006052 -id:A006052 - cp's OEIS frontend

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.

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

Gus Wiseman, Jan 13 2019

nonn

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

Chai Wah Wu, Table of n, a(n) for n = 0..59

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

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}]

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

A270876 Number of magic tori of order n composed of the numbers from 1 to n^2.

Original entry on oeis.org

1, 0, 1, 255, 251449712

Offset: 1

William Walkington, Mar 24 2016

Initially based on empirical observations by William Walkington, the results for the orders 1 to 4 have since been computed and confirmed by Walter Trump. The results for the order 5 have been computed by Walter Trump.

Dwane Campbell, Analysis of order-4 magic squares, (2013).
Dwane Campbell, Order-4 squares grouped by base square quartets, (2013).
Dwane Campbell, Features in order-4 magic squares, (2013).
William Walkington, 255 tores magiques d'ordre 4, et 1 tore magique d'ordre 3, (2011).
William Walkington, Passage du carré au tore magique, (2011).
William Walkington, 255 fourth-order magic tori, and 1 third-order magic torus, (2012).
William Walkington, From the magic square to the magic torus, (2012).
William Walkington, (using findings computed by _Walter Trump_), 251 449 712 fifth-order magic tori, (2012).
William Walkington, A new census of fourth-order magic squares, (2012).
William Walkington, Table of fourth-order magic tori, (2012).
William Walkington, 440 torus-opposite pairs of the 880 Frénicle magic squares of order-4, (2023).

Cf. A006052, A271103, A271104.

A271104 Number of magic and semi-magic tori of order n composed of the numbers from 1 to n^2.

Original entry on oeis.org

1, 0, 1, 4293, 23161722048, 2627518340149999905600

Offset: 1

William Walkington, Mar 30 2016

Initially based on empirical observations by the author, the results for the magic tori of orders 1 to 4, have since been computed and confirmed by Walter Trump. The results for the magic tori of order 5, and for the semi-magic tori of orders 4 and 5, have been computed by Walter Trump. The result for the order 6 is deduced from Artem Ripatti's findings (cf. A271103).

A semi-magic torus differs from a magic torus in that there are no magic intersections of magic diagonals, and in consequence only semi-magic squares are displayed on its surface.

Dwane Campbell, Analysis of order-4 magic squares, (2013).
Dwane Campbell, Order-4 squares grouped by base square quartets, (2013).
Dwane Campbell, Features in order-4 magic squares, (2013).
William Walkington, 255 tores magiques d'ordre 4, et 1 tore magique d'ordre 3, (2011).
William Walkington, Passage du carré au tore magique, (2011).
William Walkington, 255 fourth-order magic tori, and 1 third-order magic torus, (2012).
William Walkington, From the magic square to the magic torus, (2012).
William Walkington, (using findings computed by _Walter Trump_), 251 449 712 fifth-order magic tori, (2012).
William Walkington, A new census of fourth-order magic squares, (2012).
William Walkington, Table of fourth-order magic tori, (2012).

Cf. A006052, A270876, A271103.

a(n) = A271103(n)/ n^2.

a(6) added by William Walkington, Jul 18 2018

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

A323523 Number of positive integer square matrices with entries summing to n and equal row and column sums.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 12, 1, 7, 22, 9, 1, 64, 1, 34, 121, 11, 1, 525, 2, 13, 407, 2022, 1, 801, 1, 10163, 1036, 17, 6211, 41735, 1, 19, 2212, 285784, 1, 3822, 1, 381446, 2229142, 23, 1, 1189540, 2, 22069276, 7261, 2309410, 1, 20943183, 164176641

Offset: 0

Gus Wiseman, Jan 17 2019

nonn

Also the number of non-normal semi-magic squares with positive integer entries summing to n.

			The a(12) = 12 matrices:
  [12]
.
  [1 5] [5 1] [2 4] [4 2] [3 3]
  [5 1] [1 5] [4 2] [2 4] [3 3]
.
  [1 1 2] [1 1 2] [1 2 1] [1 2 1] [2 1 1] [2 1 1]
  [1 2 1] [2 1 1] [1 1 2] [2 1 1] [1 1 2] [1 2 1]
  [2 1 1] [1 2 1] [2 1 1] [1 1 2] [1 2 1] [1 1 2]

Andrew Howroyd, Table of n, a(n) for n = 0..200 (terms 0..59 from Chai Wah Wu)

Cf. A000290, A006052, A007016, A103198, A120732, A257493, A321719, A321722, A323349, A323523, A323524, A323529.

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]]}]];
ptnsqrs[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),And[SameQ@@Length/@#,Length[#]==0||Length[#]==Length[First[#]]]&];
Table[Sum[Length[Select[ptnsqrs[Times@@Prime/@y],And[SameQ@@Total/@#,SameQ@@Total/@Transpose[#]]&]],{y,IntegerPartitions[n]}],{n,10}]

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

a(n) = Sum_{d|n, d<=n/d} A257493(d, n/d-d) for n > 0. - Andrew Howroyd, Apr 10 2020

a(16)-a(55) from Chai Wah Wu, Jan 20 2019

A027567 Number of distinct (modulo rotation and reflection) n X n panmagic = pandiagonal = diabolic = Nasik squares.

Original entry on oeis.org

1, 0, 0, 48, 3600, 0

Offset: 1

Eric W. Weisstein

Hunter, J. A. H. and Madachy, J. S. "Mystic Arrays." Ch. 3 in Mathematical Diversions. New York: Dover, pp. 24-25, 1975.

Harvey Heinz, Pandiagonal 5 X 5.
D. N. Lehmer, A census of squares of order 4, magic in rows, columns, and diagonals, Bull. Amer. Math. Soc. 39 (1933), 981-982.
Wolfgang Müller, Group Actions on Magic Squares, Séminaire Lotharingien de Combinatoire, B39b (1997), 14 pp.
Barkley Rosser and R. J. Walker, On the transformation group for diabolic magic squares of order four, Bull. Amer. Math. Soc. 44 (1938), 416-420.
Walter Trump, How many magic squares are there? - Results of historical and computer enumeration.
Eric Weisstein's World of Mathematics, Panmagic Square
Index entries for sequences related to magic squares

Cf. A006052.

Corrected by Eric Weisstein, Mar 14 2003 to include only distinct squares; Hunter and Madachy give the count of all such squares (there are 384).

A081262 Number of distinct (modulo rotation and reflection) n X n associative magic squares.

Original entry on oeis.org

1, 0, 1, 48, 48544, 0, 1125154039419854784

Offset: 1

Eric W. Weisstein, Mar 14 2003

Go Kato, Shin-ichi Minato, Enumeration of associative magic squares of order 7, arXiv:1906.07461 [math.CO], 2019.
Walter Trump, How many magic squares are there? - Results of historical and computer enumeration.
Eric Weisstein's World of Mathematics, Associative Magic Square
Index entries for sequences related to magic squares

Cf. A006052, A081263.

a(5) from Jud McCranie, Mar 17 2003
a(6) from Max Alekseyev, May 22 2008
a(7) from Go Kato, Nov 30 2018

A081263 Number of distinct (modulo rotation and reflection) n X n associative panmagic squares.

Original entry on oeis.org

1, 0, 0, 0, 16, 0, 20190684

Offset: 1

Eric W. Weisstein, Mar 14 2003

It is known that a(10)=0.

Walter Trump estimates a(8) ~= 4.677*10^15 and a(9) ~= 1.363*10^24.

Gardner, M. "Magic Squares and Cubes." Ch. 17 in Time Travel and Other Mathematical Bewilderments. New York: W.H. Freeman, pp. 213-225, 1988.

Walter Trump, How many magic squares are there? - Results of historical and computer enumeration.
Eric Weisstein's World of Mathematics, Associative Magic Square
Index entries for sequences related to magic squares

Cf. A006052, A081262.

a(6)-a(7) from Walter Trump's website, added by Max Alekseyev, May 22 2008

A321734 Number of nonnegative integer square matrices with sum of entries equal to n, no zero rows or columns, weakly decreasing row and column sums, and the same row sums as column sums.

Original entry on oeis.org

1, 1, 3, 9, 37, 177, 1054, 7237, 57447, 512664, 5101453, 55870885, 668438484, 8667987140, 121123281293, 1814038728900, 28988885491655, 492308367375189, 8854101716492463, 168108959387012804, 3360171602215686668, 70527588239926854144, 1550926052235372201700

Offset: 0

Gus Wiseman, Nov 18 2018

nonn

			The a(3) = 9 matrices:
  [3]
.
  [2 0] [1 1]
  [0 1] [1 0]
.
  [1 0 0] [1 0 0] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
  [0 1 0] [0 0 1] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
  [0 0 1] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]

Cf. A000700, A006052, A007016, A007716, A120732, A319056, A319616, A320451, A321719, A321722, A321732, A321735, A321736, A321739.

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[Max@@First/@#]==Union[Last/@#],OrderedQ[Total/@prs2mat[#]],OrderedQ[Total/@Transpose[prs2mat[#]]],Total/@prs2mat[#]==Total/@Transpose[prs2mat[#]]]&]],{n,5}]

Let c(y) be the coefficient of m(y) in h(y), where m is monomial symmetric functions and h is homogeneous symmetric functions. Then a(n) = Sum_{|y| = n} c(y).

a(11) - a(22) from Ludovic Schwob, Sep 29 2023

A355119 a(n) is the number of order-n magic triangles composed of the numbers from 1 to n(n+1)/2 in which the sum of the k-th row and the (n-k+1)-st row is the same for all k and all three directions, counted up to rotations and reflections.

Original entry on oeis.org

1, 1, 0, 0, 7584, 5546793216

Offset: 1

Donghwi Park, Jun 19 2022

The magic sum is (n(n+1)/2 + 1)(n+1)/2.
For n >= 3, a(n) is a multiple of 6 because the rotation of only three corners does not affect the sum of the 1st row and n-th row.
This magic triangle is an analog of magic triangles from St. Olaf College, which are published in the Pi Mu Epsilon Journal (Fall 2021). Their magic triangles use square numbers of triangles.

			a(1) and a(2) are trivially 1.
a(3) is trivially 0 because the sum of 2nd row cannot be same for each direction.
a(4k) for positive integers k is trivially 0 because the magic sums are not integers in this cases.
An example of a solution at n=5:
         4
       7   9
     12  1  11
   14  2   3  13
  6  15  10  8  5
An example of a solution at n=6:
          9
        20 18
      21  8  13
    11   3  2  19
   10  6  4  7   12
 1  16  17 15  14  5

Gabriel Hale, Bjorn Vogen, and Matthew Wright, Magic Triangles, The Pi Mu Epsilon Journal (Fall 2021).
Donghwi Park, Source code for a(5)
Donghwi Park, Source code for a(6)

Cf. A000217, A006052, A008586, A342467, A356643, A356808.

a(n) = 0 if n is a multiple of 4. - Stefano Spezia, Jun 20 2022

a(6) from Donghwi Park, Dec 31 2023

cp's OEIS Frontend

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.

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A270876 Number of magic tori of order n composed of the numbers from 1 to n^2.

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A271104 Number of magic and semi-magic tori of order n composed of the numbers from 1 to n^2.

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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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A323523 Number of positive integer square matrices with entries summing to n and equal row and column sums.

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A027567 Number of distinct (modulo rotation and reflection) n X n panmagic = pandiagonal = diabolic = Nasik squares.

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A081262 Number of distinct (modulo rotation and reflection) n X n associative magic squares.

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A081263 Number of distinct (modulo rotation and reflection) n X n associative panmagic squares.

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A321734 Number of nonnegative integer square matrices with sum of entries equal to n, no zero rows or columns, weakly decreasing row and column sums, and the same row sums as column sums.

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