A336186 -id:A336186 - cp's OEIS frontend

A336189 The perfect square integer sum of a square block of integers, with 1 at the top-left corner, on a diagonally numbered 2D board.

1, 48841, 151757761, 7148452448281, 22211509021338121, 1046258952151234702321, 3250912043200499426917081, 153132136343696050161247674961, 475808694603918281112156880430641

Offset: 1

Scott R. Shannon and Eric Angelini, Jul 11 2020

See A336186 for the corresponding square edge length and an explanation of the sequence. Note that the currently known terms all end in 1.

			a(1) = 1 = 1^2.
a(2) = 48841 = 221^2.
a(3) = 151757761 = 12319^2.
a(4) = 7148452448281 = 2673659^2.
a(5) = 22211509021338121 = 149035261^2.
a(6) = 1046258952151234702321 = 32345926361^2.
a(7) = 3250912043200499426917081 = 1803028575259^2.
a(8) = 153132136343696050161247674961 = 391321014441719^2.
a(9) = 475808694603918281112156880430641 = 21813039554448121^2.
See A336186 for a diagram of the 2D board and examples.

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]

Cf. A336186, A185505, A000290, A000124, A000217.

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.

Original entry on oeis.org

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

Eric Angelini and Scott R. Shannon, Jul 13 2020

nonn

			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.

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]

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

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

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.

cp's OEIS Frontend

A336189 The perfect square integer sum of a square block of integers, with 1 at the top-left corner, on a diagonally numbered 2D board.

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