A268855 -id:A268855 - cp's OEIS frontend

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.

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.

A375459 a(n) = smallest magic sum of any 3 X 3 magic square which contains exactly n squared primes, or 0 if no such magic square exists.

Original entry on oeis.org

21, 18, 15, 51, 195, 435, 76035

Offset: 0

Arkadiusz Wesolowski, Aug 15 2024

a(9) = 0.
a(n) = A268855(n+1) for n = 0, 1, 2, and 5.
a(7) and a(8) - if nonzero, they are greater than 10^10.

Carlos Rivera, Puzzle 79. The Chebrakov's - by two months - challenge, The Prime Puzzles and Problems Connection.
Arkadiusz Wesolowski, Examples of these magic squares
Index entries for sequences related to magic squares

Cf. A001248, A268855, A375361.

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

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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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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A375459 a(n) = smallest magic sum of any 3 X 3 magic square which contains exactly n squared primes, or 0 if no such magic square exists.

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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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