A221669 -id:A221669 - 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.

A221670 Seven numbers whose squares are among the entries in a 3 x 3 magic square.

Original entry on oeis.org

23, 205, 289, 373, 425, 527, 565

Offset: 1

Jonathan Sondow, Jan 21 2013

The entries of the magic square are A221669. See there for references, links, formulas, and additional comments.

Cf. A221669.

A267118 Lee Sallows's 3 X 3 semimagic square of squares, read by rows.

Original entry on oeis.org

16129, 2116, 3364, 4, 12769, 8836, 5476, 6724, 9409

Offset: 1

Arkadiusz Wesolowski, Jan 11 2016

Three rows, three columns and one diagonal sum to the same number: 21609 = 147^2.

See the link to mersenneforum.org about triangular numbers that form such semimagic squares.

			The semimagic square is
|-----|-----|-----|
|16129| 2116| 3364|
|-----|-----|-----|
|  4  |12769| 8836|
|-----|-----|-----|
| 5476| 6724| 9409|
|-----|-----|-----|
It is:
|-----|-----|-----|
|127^2| 46^2| 58^2|
|-----|-----|-----|
| 2^2 |113^2| 94^2|
|-----|-----|-----|
| 74^2| 82^2| 97^2|
|-----|-----|-----|

Christian Boyer, Magic squares of squares
mersenneforum.org, discussion titled A special semimagic square
Carlos Rivera, Problem 63
Lee Sallows, The lost theorem, The Mathematical Intelligencer, 19:4 (1997), pp. 51-54.
Index entries for sequences related to magic squares

Cf. A000290, A221669.

A319589 A 3 X 3 magic square with five square entries, read by rows.

Original entry on oeis.org

34969, 83521, 7585, 14641, 42025, 69409, 76465, 529, 49081

Offset: 1

Arkadiusz Wesolowski, Sep 23 2018

Some magic squares of order 3 with five square entries have this parametric form:
|-----------------------|-----------------------|-----------------------|
|        (x*y)^2        |          y^4          |sqrt(z)*(3*x^2 - y^2)/2|
|-----------------------|-----------------------|-----------------------|
|          x^4          |           z           |       2*z - x^4       |
|-----------------------|-----------------------|-----------------------|
|sqrt(z)*(3*y^2 - x^2)/2|       2*z - y^4       |     2*z - (x*y)^2     |
|-----------------------|-----------------------|-----------------------|
where z = (x^2 + y^2)^2/4, x and y are integers such that (x^4 - y^4 + 2*(x*y)^2)/2 is a square (e.g., x = 11 and y = 17; x = 5337 and y = 6257).
This sequence presents the magic square belonging to this family and having the smallest possible magic sum (S = 126075).

			The magic square is
+-------+-------+-------+
| 187^2 |  17^4 |  7585 |
+-------+-------+-------+
|  11^4 | 205^2 | 69409 |
+-------+-------+-------+
| 76465 |  23^2 | 49081 |
+-------+-------+-------+

Index entries for sequences related to magic squares

Cf. A221669, A268855.

A375361 Odd numbers with at least two prime divisors of the form 4*k + 1 counted with multiplicity.

Original entry on oeis.org

25, 65, 75, 85, 125, 145, 169, 175, 185, 195, 205, 221, 225, 255, 265, 275, 289, 305, 325, 365, 375, 377, 425, 435, 445, 455, 475, 481, 485, 493, 505, 507, 525, 533, 545, 555, 565, 575, 585, 595, 615, 625, 629, 663, 675, 685, 689, 697, 715, 725, 745, 765, 775

Offset: 1

Arkadiusz Wesolowski, Aug 13 2024

nonn

Odd numbers k such that k^2 can be expressed as the arithmetic mean of two distinct perfect squares in more than one way. For example, 25^2 = (5^2 + 35^2)/2 = (17^2 + 31^2)/2.

Let x be a squared integer which is the central element of a 3 X 3 magic square in which seven (or more) of the entries are squared integers. If the greatest common divisor of all nine entries is 1, then the square root of x is a composite number that is divisible only by primes congruent to 1 mod 4. For example, sqrt(A221669(5)) = 425 is both in A004613 and in this sequence.

			65 is in this sequence because 65 has two prime factors of the form 4*k + 1, namely 5 = 4*1 + 1 and 13 = 4*3 + 1.

Cf. A004613, A008844, A221669, A221670, A268855, A309812, A375459.

f:=func; nopf:=func; sum:=func; [n: n in [1..775 by 2] | sum(n) gt 1];

isok(n) = my(v=Vec(factor(n))); n%2&&sum(t=1, omega(n), if((v[1]%4)[t]==1, v[2][t]))>1;

isok(n) = my(t); if(n%2, for(k=sqrtint(n^2-1)+2, sqrtint(2*n^2-1), if(issquare(2*n^2-k^2)&&t++>1, return(1)))); 0;

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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A221670 Seven numbers whose squares are among the entries in a 3 x 3 magic square.

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A267118 Lee Sallows's 3 X 3 semimagic square of squares, read by rows.

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A319589 A 3 X 3 magic square with five square entries, read by rows.

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A375361 Odd numbers with at least two prime divisors of the form 4*k + 1 counted with multiplicity.

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