A221670 -id:A221670 - cp's OEIS frontend

A221669 3 x 3 magic square containing seven squares, read by rows.

139129, 83521, 319225, 360721, 180625, 529, 42025, 277729, 222121

Offset: 1

Jonathan Sondow, Jan 21 2013

Apart from trivial modifications, this is the only known 3 x 3 magic square containing seven squares. No 3 x 3 magic square containing eight or nine squares is known.

The seven integer square roots are A221670.

			[139129 83521 319225
360721 180625 529
42025 277729 222121]

A. Bremner, On squares of squares, Acta Arith. 88 (1999), 289-297.
A. Bremner, On squares of squares. II, Acta Arith. 99 (2001), 289-308.
John P. Robertson, Magic squares of squares, Math. Mag., 69 (1996), 289-293.
Zentralblatt, Review of "On squares of squares. II" by A. Bremner
Index entries for sequences related to magic squares

Cf. A221670, A319589.

a(1) = 139129 = 373^2, a(2) = 83521 = 289^2, a(3) = 319225 = 565^2, a(5) = 180625 = 425^2, a(6) = 529 = 23^2, a(7) = 42025 = 205^2, a(8) = 277729 = 527^2.

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

A221669 3 x 3 magic square containing seven squares, read by rows.

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