A338642 Square spiral of smallest distinct positive integers starting at 1 such that the four sums of each term with its four nearest neighbors are composite numbers.

1, 3, 5, 7, 2, 8, 4, 11, 9, 6, 12, 10, 14, 13, 15, 18, 16, 19, 17, 21, 23, 22, 24, 25, 20, 26, 28, 27, 29, 31, 32, 30, 33, 35, 34, 36, 38, 39, 37, 40, 41, 43, 42, 45, 46, 44, 47, 48, 50, 49, 51, 53, 55, 56, 54, 52, 58, 59, 57, 60, 61, 62, 63, 66, 64, 68, 65, 67, 71, 69, 72, 70, 73, 74, 76, 77, 75

Offset: 1

Author

Eric Angelini and Scott R. Shannon, Apr 21 2021

Keywords

• nonn

Examples

			The square spiral starts:
.
     38--36--34--35--33--30--32
      |                       |
     39  16--18--15--13--14  31
      |   |               |   |
     37  19   2---7---5  10  29
      |   |   |       |   |   |
     40  17   8   1---3  12  27
      |   |   |           |   |
     41  21   4--11---9---6  28
      |   |                   |
     43  23--22--24--25--20--26
      |
     42--45--46--44--47--48--50..
.
a(2) = 3 as a(1) + 3 = 1 + 3 = 4, the smallest possible composite number.
a(3) = 5 as a(2) + 5 = 3 + 5 = 8. Note a(3) cannot be 2 or 4 as when these are added to 3 the result is a prime number.
a(4) = 7 as a(3) + 7 = 5 + 7 = 12, and a(1) + 7 = 1 + 7 = 8, both being composite.
a(9) = 9 as a(8) + 9 = 11 + 9 = 20, and a(2) + 9 = 3 + 9 = 12, both being composite.
		

Crossrefs

Cf. A338644 (sum to primes), A002808, A063826, A260643, A334742, A307834, A338221.