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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.

%I A336335 #14 Jul 25 2020 17:09:58
%S A336335 11,28,50,176,452,536,848,1388,2048,1682,3752,4784,6272,7268,8696,
%T A336335 7938,13748,14210,17756,19952,11888,24728,27308,25322,20456,38888,
%U A336335 42128,45476,32792,49826,64136,68252,43698,76868,77930,90752,69216,105788,111056,108354,127628
%N 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.
%H A336335 Hugo Pfoertner, <a href="/A336335/b336335.txt">Table of n, a(n) for n = 1..1000</a>
%F A336335 a(n) = A054552(prime(n)) if prime(n) != 1 mod 4.
%e A336335   37--36--35--34--33--32--31
%e A336335    |                       |
%e A336335   38  17--16--15--14--13  30  ...
%e A336335    |   |               |   |   |
%e A336335   39  18   5---4---3  12  29  54
%e A336335    |   |   |       |   |   |   |
%e A336335   40  19   6   1---2 d=2 d=3  53
%e A336335    |   |   |           |   |   |
%e A336335   41  20   7---8---9--10  27  52
%e A336335    |   |                   |   |
%e A336335   42  21--22--23--24--25--26  51
%e A336335    |                           |
%e A336335   43--44--45--46--47--48--49-d=5
%e A336335 .
%e A336335 a(1) = 11 is the index of the first occurrence of distance d = 2 = prime(1) from the start of the spiral.
%e A336335 a(2) = 28 is the index of the first occurrence of distance d = 3 = prime(2) from the start of the spiral.
%e A336335 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.
%Y A336335 Cf. A002144, A002145, A054552, A174344, A268038, A274923, A336336.
%K A336335 nonn
%O A336335 1,1
%A A336335 _Hugo Pfoertner_, Jul 24 2020