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

1,2

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]

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.

Example

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.

Mathematica

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

Crossrefs

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

Sequence in context: A032678 A073671 A172367 * A024901 A258011 A000959

Adjacent sequences: A336231 A336232 A336233 * A336235 A336236 A336237

Keyword

nonn

Author

Eric Angelini and Scott R. Shannon, Jul 13 2020

Status

approved