A336335 a(n) is the index of the first occurrence of the Euclidean distance prime(n) from a point on a square spiral to its starting point at 1.

11, 28, 50, 176, 452, 536, 848, 1388, 2048, 1682, 3752, 4784, 6272, 7268, 8696, 7938, 13748, 14210, 17756, 19952, 11888, 24728, 27308, 25322, 20456, 38888, 42128, 45476, 32792, 49826, 64136, 68252, 43698, 76868, 77930, 90752, 69216, 105788, 111056, 108354, 127628

Offset: 1

Author

Hugo Pfoertner, Jul 24 2020

Keywords

• nonn

Examples

			  37--36--35--34--33--32--31
   |                       |
  38  17--16--15--14--13  30  ...
   |   |               |   |   |
  39  18   5---4---3  12  29  54
   |   |   |       |   |   |   |
  40  19   6   1---2 d=2 d=3  53
   |   |   |           |   |   |
  41  20   7---8---9--10  27  52
   |   |                   |   |
  42  21--22--23--24--25--26  51
   |                           |
  43--44--45--46--47--48--49-d=5
.
a(1) = 11 is the index of the first occurrence of distance d = 2 = prime(1) from the start of the spiral.
a(2) = 28 is the index of the first occurrence of distance d = 3 = prime(2) from the start of the spiral.
Distances of the form 4*k+1 corresponding to Pythagorean primes A002144 occur earlier than on the East spoke of the square spiral, dependent on the decomposition of p^2 into two squares. prime(3)^2 = 4^2 + 3^2 leads to index a(3) = 50 in the spiral.
		

Links

• Hugo Pfoertner, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A002144, A002145, A054552, A174344, A268038, A274923, A336336.

Formula

a(n) = A054552(prime(n)) if prime(n) != 1 mod 4.