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

1,1

Links

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

Formula

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

Example

  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.

Crossrefs

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

Sequence in context: A065006 A291058 A061952 * A140677 A264443 A364893

Adjacent sequences: A336332 A336333 A336334 * A336336 A336337 A336338

Keyword

nonn

Author

Hugo Pfoertner, Jul 24 2020

Status

approved