A336234 Edge length of 'Prime squares': sum the four numbers at the corners of a square drawn on a diagonally numbered 2D board, with 1 at the corner of the square. The sequence gives the size of the square such that the sum is a prime number.

1, 3, 7, 9, 13, 15, 19, 25, 31, 37, 39, 51, 61, 63, 69, 81, 87, 97, 99, 109, 117, 135, 145, 147, 151, 153, 163, 165, 171, 183, 189, 195, 201, 207, 213, 219, 223, 229, 235, 241, 249, 253, 267, 271, 273, 277, 297, 307, 319, 325, 337, 343, 345, 355, 373, 381, 387, 391, 393, 409, 435, 447, 451, 457

Offset: 1

Author

Eric Angelini and Scott R. Shannon, Jul 13 2020

Keywords

• nonn

Examples

			The board is numbered as follows:
.
   1  2  4  7 11 16  .
   3  5  8 12 17  .
   6  9 13 18  .
  10 14 19  .
  15 20  .
  21  .
  .
a(1) = 1 as the four numbers {1,2,5,3} form the corners of a square of size 1, and the sum of these number is 11, a prime number.
a(2) = 3 as the four numbers {1,7,25,10} form the corners of a square of size 3, and the sum of these number is 43, a prime number.
a(3) = 7 as the four numbers {1,29,113,36} form the corners of a square of size 7, and the sum of these number is 179, a prime number.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..2000
• Eric Angelini, Prime squares and square squares, personal blog "Cinquante signes", Jun. 29, 2020.
• Eric Angelini, Prime squares and square squares, personal blog "Cinquante signes", Jun. 29, 2020. [Cached copy]

Crossrefs

Cf. A000040, A185505, A000124, A000217, A282513.

Programs

Mathematica

Select[Range[1,501,2],PrimeQ[3#^2+4#+4]&] (* Harvey P. Dale, May 26 2022 *)

Formula

The sequence is the values of d where 3*d^2+4*d+4, the sum of the four numbers for a square of size d, is prime. For even d this sum will always be even, thus all terms are odd.