A157182 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.

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

Offset

1,2

Comments

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?

Links

Table of n, a(n) for n=1..72.

D. Wilson and F. Adams-Watters in reply to Leroy Quet, Prime Sums In A Grid, SeqFan list, Feb 24 2009

Example

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

Prog

(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 ]; }

Crossrefs

This is the 2D analog of A055265. [From Franklin T. Adams-Watters, Mar 07 2009]

Sequence in context: A356222 A329901 A284572 * A343232 A292145 A297552

Adjacent sequences: A157179 A157180 A157181 * A157183 A157184 A157185

Keyword

nonn,tabl,changed

Author

M. F. Hasler, Feb 24 2009

Status

approved