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A132163 Triangle read by rows. For row n, start with 1 but from the second term onwards always choose the largest positive integer between 1 and n inclusive that i) has not already appeared in the row ii) gives a prime when added to the previous term. Stop if no such integer can be found.

%I A132163 #13 Nov 26 2015 17:13:14
%S A132163 1,1,2,1,2,3,1,4,3,2,1,4,3,2,5,1,6,5,2,3,4,1,6,7,4,3,2,5,1,6,7,4,3,8,
%T A132163 5,2,1,6,7,4,9,8,5,2,3,1,10,9,8,5,6,7,4,3,2,1,10,9,8,11,6,7,4,3,2,5,1,
%U A132163 12,11,8,9,10,7,6,5,2,3,4
%N A132163 Triangle read by rows. For row n, start with 1 but from the second term onwards always choose the largest positive integer between 1 and n inclusive that i) has not already appeared in the row ii) gives a prime when added to the previous term. Stop if no such integer can be found.
%C A132163 The following statements are conjectural: 1) The n-th row is always a permutation of 1,...,n. 2) For the even rows, the last term is one less than a prime (so the row gives a solution to the prime circle problem - see A051252). 3) There exists a (unique) sequence b(2), b(3),... with the property that for every n > 1 there is a positive integer N such that every even row of the triangle from the 2N-th onwards ends b(n), ..., b(3), b(2) and every odd row from the (2N - 1)-th onwards ends b(n)+(-1)^n, ..., b(3)-1, b(2)+1. (If the sequence b(n) exists it is probably A132075 without the initial term 1.)
%H A132163 Reinhard Zumkeller, <a href="/A132163/b132163.txt">Rows n = 1..150 of triangle, flattened</a>
%t A132163 t[_, 1] = 1; t[n_, k_] := t[n, k] = For[ j = n, j > 1, j--, If[ PrimeQ[ t[n, k-1] + j] && FreeQ[ Table[ t[n, m], {m, 1, k-1}], j], Return[j] ] ]; Table[ t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Apr 02 2013 *)
%o A132163 (Haskell)
%o A132163 import Data.List (delete)
%o A132163 a132163_tabl = map a132163_row [1..]
%o A132163 a132163 n k = a132163_row n !! (k-1)
%o A132163 a132163_row n = 1 : f 1 [n, n-1 .. 2] where
%o A132163    f u vs = g vs where
%o A132163      g []                            = []
%o A132163      g (x:xs) | a010051 (x + u) == 1 = x : f x (delete x vs)
%o A132163               | otherwise            = g xs
%o A132163 -- _Reinhard Zumkeller_, Jan 05 2013
%Y A132163 This sequence is a variation on A088643.
%K A132163 easy,nice,nonn,tabl
%O A132163 1,3
%A A132163 _Paul Boddington_, Nov 04 2007