cp's OEIS Frontend

A353591 Lexicographically earliest permutation of the nonnegative integers filling an infinite square array by falling antidiagonals so that the elements on any 2 X 2 square sum to a prime.

%I A353591 #10 Jun 14 2022 06:57:11
%S A353591 0,1,2,3,4,5,6,9,8,7,10,11,16,17,12,13,14,23,18,25,15,19,22,31,26,29,
%T A353591 21,20,24,35,30,27,28,32,33,34,36,49,40,39,46,38,41,44,37,42,48,43,54,
%U A353591 45,51,52,55,58,47,50,53,57,56,59,69,70,63,66,81,60,61,78,65,67,64,68,73,72,79,76
%N A353591 Lexicographically earliest permutation of the nonnegative integers filling an infinite square array by falling antidiagonals so that the elements on any 2 X 2 square sum to a prime.
%C A353591 In A337116 the infinite 2D lattice is filled along a square spiral satisfying the same constraint of 2 X 2 squares adding up to primes.
%e A353591 The square array starts
%e A353591    0   1   3   6  10  13  19  24  36  42  ...
%e A353591    2   4   9  11  14  22  35  49  48  ...
%e A353591    5   8  16  23  31  30  40  43  ...
%e A353591    7  17  18  26  27  39  54  ...
%e A353591   12  25  29  28  46  45  ...
%e A353591   15  21  32  38  51  ...
%e A353591   20  33  41  52  ...
%e A353591   34  44  55  ...
%e A353591   37  58  ...
%e A353591   47  ...
%e A353591   ...
%e A353591 a(4) is in the second row and column. It must sum up with a(0) = 0, a(1) = 1 and a(2) = 2 to a prime. The smallest possible solution is to reach the prime p = 7 with a(4) = 4.
%e A353591 Similarly, a(7) which is on the second row, third column, must sum up with a(1) = 1 (above to the left), a(3) = 3 (above) and a(4) = 4 (to the left) to a prime; the smallest solution is to reach the prime p = 17 using a(7) = 9.
%o A353591 (PARI) A353591_upto(N, M=Map(), r,c, U=[-1])={vector(N, i, if(r && c, my(s=mapget(M,[r-1,c-1])+mapget(M,[r-1,c])+mapget(M,[r,c-1]), p=nextprime(s+U[1]+1)); while(setsearch(U, N=p-s), p=nextprime(p+1)), N=U[1]+1); mapput(M,[r,c], N); if(c, c--;r++, r=!c=r+1); U=setunion(U, [N]); while(#U>1 && U[2]==U[1]+1, U=U[^1]); N)}
%Y A353591 Cf. A000040 (the primes), A337116 (same idea with square spiral instead of array by antidiagonals), A353590 (same idea with squares instead of primes).
%K A353591 nonn
%O A353591 0,3
%A A353591 _M. F. Hasler_, May 29 2022