A268856 -id:A268856 - cp's OEIS frontend

A268855 a(n) = smallest magic sum of any 3 X 3 magic square which contains exactly n squares.

Original entry on oeis.org

30, 21, 18, 15, 39, 120, 435, 541875

Offset: 0

Arkadiusz Wesolowski, Feb 14 2016

a(7) = 8*A268854(7) + 3.

If a(8) exists, and the central element is a square, then a(8) > 10^21. - Arkadiusz Wesolowski, Sep 01 2024

Andrew Bremner, On squares of squares II, Acta Arithmetica 99 (2001), pp. 289-308.
Multimagic Squares, Magic squares of squares
The Prime Puzzles and Problems Connection, Problem 63
The Prime Puzzles and Problems Connection, Puzzle 79
Arkadiusz Wesolowski, Examples of these magic squares
Wikipedia, Magic square
Index entries for sequences related to magic squares

Cf. A221669, A268854, A268856, A269231, A375459.

A269231 a(n) = smallest magic sum of any 3 X 3 semimagic square which contains exactly n cubes.

Original entry on oeis.org

19, 16, 15, 32, 73, 153, 1520, 5104, 241801435

Offset: 0

Arkadiusz Wesolowski, Feb 20 2016

Multimagic Squares, Magic squares of cubes
The Prime Puzzles and Problems Connection, Puzzle 412
Arkadiusz Wesolowski, Examples of these semimagic squares
Index entries for sequences related to magic squares

Cf. A268854, A268855, A268856.

a(8) from Christian Boyer, added by Arkadiusz Wesolowski, Apr 14 2016

A268854 a(n) is the smallest magic sum of any 3 X 3 magic square which contains exactly n triangular numbers.

Original entry on oeis.org

39, 24, 21, 15, 18, 24, 189, 67734

Offset: 0

Arkadiusz Wesolowski, Feb 14 2016

Andrew Bremner, On squares of squares II, Acta Arithmetica 99 (2001), pp. 289-308.
Multimagic Squares, An unsolved problem
The Prime Puzzles and Problems Connection, Problem 63
The Prime Puzzles and Problems Connection, Puzzle 79
Arkadiusz Wesolowski, Examples of these magic squares
Wikipedia, Magic square
Index entries for sequences related to magic squares

Cf. A221669, A268855, A268856, A269231, A271020.

A368339 Take the solution to Pellian equation x^2 - 8ny^2 = 1 with smallest positive y and x >= 0; sequence gives a(n) = y, or 0 if n is twice a positive square. A368340 gives values of x.

Original entry on oeis.org

1, 0, 1, 3, 3, 1, 2, 0, 2, 1, 21, 5, 5, 12, 1, 51, 3, 0, 3, 57, 1, 15, 1794, 7, 7, 45, 33, 1, 1287, 2, 4, 0, 4, 2, 15, 1, 215, 3315, 3, 9, 9, 3, 561, 4137, 1, 60, 110532, 245, 5, 0, 5, 255, 1557945, 65, 1, 6, 48, 455, 14127, 11, 11, 207480, 20, 29427, 285, 1

Offset: 1

Arkadiusz Wesolowski, Dec 21 2023

			For n = 1, 2, 3, 4, 5 solutions are (x,y) = (3, 1), (1, 0), (5, 1), (17, 3), (19, 3).

Carlos Rivera, Problem 88. Follow-up to problem 63, The Prime Puzzles & Problems Connection.

Cf. A000217, A002349, A268856, A368340.

pellsolve(n)={if(issquare(n/2), return(0), q=bnfinit('x^2-8*n, 1); i=-1; until(y&&x==floor(x)&&y==floor(y)&&x^2-8*n*y^2==1, f=lift(q.fu[1]^i); x=abs(polcoeff(f, 0)); y=abs(polcoeff(f, 1)); i++); return(y))};

a(n) = A002349(8*n).
a(n) = sqrt((A368340(n)^2 - 1)/(8*n)).
a(A000217(n)) = 1, n >= 1.

cp's OEIS Frontend

A268855 a(n) = smallest magic sum of any 3 X 3 magic square which contains exactly n squares.

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A269231 a(n) = smallest magic sum of any 3 X 3 semimagic square which contains exactly n cubes.

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A268854 a(n) is the smallest magic sum of any 3 X 3 magic square which contains exactly n triangular numbers.

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A368339 Take the solution to Pellian equation x^2 - 8ny^2 = 1 with smallest positive y and x >= 0; sequence gives a(n) = y, or 0 if n is twice a positive square. A368340 gives values of x.

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cp's OEIS Frontend

A268855 a(n) = smallest magic sum of any 3 X 3 magic square which contains exactly n squares.

Views

Author

Keywords

Comments

Links

Crossrefs

A269231 a(n) = smallest magic sum of any 3 X 3 semimagic square which contains exactly n cubes.

Views

Author

Keywords

Links

Crossrefs

Extensions

A268854 a(n) is the smallest magic sum of any 3 X 3 magic square which contains exactly n triangular numbers.

Views

Author

Keywords

Links

Crossrefs

A368339 Take the solution to Pellian equation x^2 - 8*n*y^2 = 1 with smallest positive y and x >= 0; sequence gives a(n) = y, or 0 if n is twice a positive square. A368340 gives values of x.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A368339 Take the solution to Pellian equation x^2 - 8ny^2 = 1 with smallest positive y and x >= 0; sequence gives a(n) = y, or 0 if n is twice a positive square. A368340 gives values of x.