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A157182
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Square array such that horizontal and vertical neighbors add to a prime, read by antidiagonals, filled the greedy way using positive integers, not more than once each.
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1
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1, 2, 4, 3, 9, 7, 8, 10, 22, 6, 5, 21, 19, 25, 11, 12, 26, 40, 18, 36, 20, 17, 35, 27, 13, 23, 47, 33, 14, 24, 32, 16, 30, 50, 56, 28, 15, 29, 65, 51, 31, 53, 57, 45, 39, 38, 44, 42, 62, 52, 48, 74, 82, 34, 58, 41, 59, 95, 89, 75, 49, 83, 99, 55, 69, 43, 60, 68, 54, 78, 92, 64
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OFFSET
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1,2
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COMMENTS
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Following ideas from Leroy Quet, D. Wilson and F. Adams-Watters, cf. link.
Can we conjecture that this is a permutation of the positive integers?
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LINKS
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D. Wilson and F. Adams-Watters in reply to _Leroy Quet_, Prime Sums In A Grid, SeqFan list, Feb 24 2009
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EXAMPLE
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The antidiagonals of the array form the following triangle, where each number must add up to a prime with its neighbors above:
____________________________ 1,
__________________________ 2 , 4,
________________________ 3 , 9 , 7,
______________________ 8, 10 , 22 , 6,
____________________ 5, 21 , 19 , 25 , 11,
_________________ 12, 26 , 40 , 18 , 36 , 20,
_______________ 17, 35 , 27 , 13 , 23 , 47 , 33,
_____________ 14, 24 , 32 , 16 , 30 , 50 , 56 , 28,
___________ 15, 29 , 65 , 51 , 31 , 53 , 57 , 45 , 39,
_________ 38, 44 , 42 , 62 , 52 , 48 , 74 , 82 , 34 , 58,
_______ 41, 59 , 95 , 89 , 75 , 49 , 83 , 99 , 55 , 69 , 43,
_____ 60, 68 , 54 , 78 , 92 , 64 , 90, 80 , 94 , 112, 70 , 46,
___ 37, 71 , 113, 73, 101 , 87 , 67, 77, 117 , 79 , 61 , 81 , 63,
_ 66, 102, 86 , 84, 126, 110, 106, 72, 116, 154 , 88 , 76, 100, 104,
85, 91, 125, 107, 97, 131, 123, 121, 155, 153, 103, 105, 151, 93, 119
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PROG
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(PARI) A157182( n, show=0/*set to 1 to print everything instead*/, last_diag=[1], min_not_used=2, others_used=[])={ local(new_diag); n-- | return(1); for( d=1+#last_diag, 1+sqrtint(2*n), /* fill the d-th antidiagonal */ show & print(last_diag, ", "); new_diag=vector( d ); for( j=1, d, ! new_diag[ j ] && new_diag[ j ] = min_not_used + (d-min_not_used)%2; while( setsearch( others_used, new_diag[j] ) | ( j > 1 && setsearch( Set( vecextract( new_diag, 2^(j-1)-1)), new_diag[j] )) | ( j < d && ! isprime( last_diag[ j ] + new_diag[ j ] )) | ( j > 1 && ! isprime( last_diag[ j-1 ] + new_diag[ j ] )), new_diag[j] += 2; ); show | n-- | return(new_diag[j]) ); others_used = setunion( others_used, new_diag ); while( setsearch( others_used, min_not_used ), others_used = setminus( others_used, Set( min_not_used )); min_not_used++; ); last_diag=new_diag; /* a=concat(a, new_diag ); */ ); [ min_not_used, others_used ]; }
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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