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A336234 Edge length of 'Prime squares': sum the four numbers at the corners of a square drawn on a diagonally numbered 2D board, with 1 at the corner of the square. The sequence gives the size of the square such that the sum is a prime number.

%I A336234 #18 Dec 21 2024 18:39:51
%S A336234 1,3,7,9,13,15,19,25,31,37,39,51,61,63,69,81,87,97,99,109,117,135,145,
%T A336234 147,151,153,163,165,171,183,189,195,201,207,213,219,223,229,235,241,
%U A336234 249,253,267,271,273,277,297,307,319,325,337,343,345,355,373,381,387,391,393,409,435,447,451,457
%N A336234 Edge length of 'Prime squares': sum the four numbers at the corners of a square drawn on a diagonally numbered 2D board, with 1 at the corner of the square. The sequence gives the size of the square such that the sum is a prime number.
%H A336234 Harvey P. Dale, <a href="/A336234/b336234.txt">Table of n, a(n) for n = 1..2000</a>
%H A336234 Eric Angelini, <a href="https://cinquantesignes.blogspot.com/2020/06/prime-squares.html">Prime squares and square squares</a>, personal blog "Cinquante signes", Jun. 29, 2020.
%H A336234 Eric Angelini, <a href="/A336186/a336186.pdf">Prime squares and square squares</a>, personal blog "Cinquante signes", Jun. 29, 2020. [Cached copy]
%F A336234 The sequence is the values of d where 3*d^2+4*d+4, the sum of the four numbers for a square of size d, is prime. For even d this sum will always be even, thus all terms are odd.
%e A336234 The board is numbered as follows:
%e A336234 .
%e A336234    1  2  4  7 11 16  .
%e A336234    3  5  8 12 17  .
%e A336234    6  9 13 18  .
%e A336234   10 14 19  .
%e A336234   15 20  .
%e A336234   21  .
%e A336234   .
%e A336234 a(1) = 1 as the four numbers {1,2,5,3} form the corners of a square of size 1, and the sum of these number is 11, a prime number.
%e A336234 a(2) = 3 as the four numbers {1,7,25,10} form the corners of a square of size 3, and the sum of these number is 43, a prime number.
%e A336234 a(3) = 7 as the four numbers {1,29,113,36} form the corners of a square of size 7, and the sum of these number is 179, a prime number.
%t A336234 Select[Range[1,501,2],PrimeQ[3#^2+4#+4]&] (* _Harvey P. Dale_, May 26 2022 *)
%Y A336234 Cf. A000040, A185505, A000124, A000217, A282513.
%K A336234 nonn
%O A336234 1,2
%A A336234 _Eric Angelini_ and _Scott R. Shannon_, Jul 13 2020