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A353591
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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.
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1
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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, 21, 20, 24, 35, 30, 27, 28, 32, 33, 34, 36, 49, 40, 39, 46, 38, 41, 44, 37, 42, 48, 43, 54, 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
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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EXAMPLE
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The square array starts
0 1 3 6 10 13 19 24 36 42 ...
2 4 9 11 14 22 35 49 48 ...
5 8 16 23 31 30 40 43 ...
7 17 18 26 27 39 54 ...
12 25 29 28 46 45 ...
15 21 32 38 51 ...
20 33 41 52 ...
34 44 55 ...
37 58 ...
47 ...
...
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.
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.
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PROG
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(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)}
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CROSSREFS
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Cf. A000040 (the primes), A337116 (same idea with square spiral instead of array by antidiagonals), A353590 (same idea with squares instead of primes).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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