A269231 -id:A269231 - 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.

A268856 a(n) = smallest magic sum of any 3 X 3 magic square which contains exactly n numbers that are triangular or square.

Original entry on oeis.org

39, 36, 30, 21, 21, 15, 27, 675

Offset: 0

Arkadiusz Wesolowski, Feb 14 2016

a(8) and a(9) are greater than 10^10. - Arkadiusz Wesolowski, Oct 27 2018

a(8) <= 3*7215727335550^2 = (A221669(1) + A221669(2) + A221669(3))*16978181966^2. - Arkadiusz Wesolowski, May 21 2022

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

Cf. A268854, A268855, A269231.

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.

A271450 a(n) = smallest magic sum of any 3 X 3 magic square which contains exactly n cubes, or 0 if no such magic square exists.

Original entry on oeis.org

27, 18, 15, 42, 882

Offset: 0

Arkadiusz Wesolowski, Apr 14 2016

For n >= 7, a(n) = 0 because the equation x^n + y^n = 2*z^n has no solution for n > 2 apart from the trivial x = y = z.

Richard K. Guy, Unsolved Problems in Number Theory, New York: Springer-Verlag, 2004, pp. 268-271.

Henri Darmon and Loic Merel, Winding quotients and some variants of Fermat’s Last Theorem
Arkadiusz Wesolowski, Examples of these magic squares
Wikipedia, Fermat's Last Theorem
Index entries for sequences related to magic squares

Cf. A269231.

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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A268856 a(n) = smallest magic sum of any 3 X 3 magic square which contains exactly n numbers that are triangular or square.

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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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A271450 a(n) = smallest magic sum of any 3 X 3 magic square which contains exactly n cubes, or 0 if no such magic square exists.

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