A344660 -id:A344660 - cp's OEIS frontend

A344659 Lexicographically earliest sequence of distinct nonnegative terms on a square spiral such that each term forms no prime value in the eight sums when each term is added to each of its eight nearest neighbors.

0, 1, 8, 14, 4, 6, 9, 15, 20, 5, 7, 13, 2, 12, 10, 11, 16, 22, 18, 3, 17, 19, 25, 29, 28, 21, 23, 27, 31, 26, 24, 30, 32, 34, 35, 33, 36, 41, 40, 42, 39, 37, 38, 43, 45, 47, 49, 53, 57, 48, 51, 54, 61, 50, 44, 46, 52, 56, 58, 60, 59, 65, 55, 62, 63, 70, 64, 74, 66, 69, 67, 68, 72, 73, 77, 71, 75

Offset: 1

Eric Angelini and Scott R. Shannon, May 26 2021

nonn

.
  36--33--35--34--32--30--24
   |                       |
  41  16--11--10--12---2  26
   |   |               |   |
  40  22   4--14---8  13  31
   |   |   |       |   |   |
  42  18   6   0---1   7  27
   |   |   |           |   |
  39   3   9--15--20---5  23   .
   |   |                   |   .
  37  17--19--25--29--28--21   .
   |                           |
  38--43--45--47--49--53--57--48
.

			The eight terms that are in contact with the initial zero are 1, 8, 14, 4, 6, 9, 15, 20: none of them is prime [forcing the sum a(k) + 0 to be nonprime, with k<9]; more generally, no term of the square spiral when added to any of its eight nearest neighbors sums to a prime.

Cf. A063826, A344660, A002808, A000040.

A354111 Lexicographically earliest sequence of distinct nonnegative terms on a square spiral such that for any 2 X 2 square of numbers both the sum of those numbers and the sum of the digits of those numbers add up to a square. Start with a(0) = 0.

Original entry on oeis.org

0, 1, 2, 6, 3, 7, 4, 5, 19, 8, 141, 25, 9, 133, 28, 132, 10, 24, 135, 23, 11, 131, 29, 91, 26, 12, 98, 378, 32, 78, 13, 44, 39, 124, 157, 230, 14, 275, 220, 105, 178, 229, 15, 69, 365, 51, 54, 153, 385, 16, 163, 303, 62, 104, 227, 123, 17, 43, 476, 66, 212, 83, 106, 134, 18, 30, 210, 195, 56

Offset: 0

Scott R. Shannon and Eric Angelini, May 17 2022

			The board is numbered with the square spiral:
.
  10--132--28--133--9   .
   |                |   .
  24   3---6---2   25   32
   |   |       |    |   |
  135  7   0---1   141 378
   |   |            |   |
  23   4---5---19---8   98
   |                    |
  11--131--29--91--26---12
.
.
0 + 1 + 2 + 6 = 9 = 3^2;
0 + 6 + 3 + 7 = 16 = 4^2;
0 + 5 + 19 + 1 = 25 = 5^2, and 0 + 5 + 1 + 9 + 1 = 16 = 4^2;
0 + 7 + 4 + 5 = 16 = 4^2;
1 + 141 + 25 + 2 = 169 = 13^2, and 1 + 1 + 4 + 1 + 2 + 5 + 2 = 16 = 4^2;
141 + 378 + 32 + 25 = 576 = 24^2, and 1 + 4 + 1 + 3 + 7 + 8 + 3 + 2 + 2 + 5 = 36 = 6^2;

Cf A337115, A347333, A344660, A344659.

cp's OEIS Frontend

A344659 Lexicographically earliest sequence of distinct nonnegative terms on a square spiral such that each term forms no prime value in the eight sums when each term is added to each of its eight nearest neighbors.

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