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A297615 Triangular array T(n, k) read by rows, n > 0, 0 < k <= n: T(n, k) = least unused positive value (reading rows from left to right) such that each triple of pairwise adjacent terms sums to a prime.

%I A297615 #16 Nov 17 2018 20:55:46
%S A297615 1,2,4,6,5,8,3,20,16,7,10,18,23,14,22,12,9,26,24,35,32,11,38,36,17,30,
%T A297615 42,15,28,34,29,44,66,31,40,46,48,21,76,58,47,54,78,45,60,13,70,82,33,
%U A297615 88,56,41,74,62,27,50,68,59,52,72,19,136,64,43,92,80,84
%N A297615 Triangular array T(n, k) read by rows, n > 0, 0 < k <= n: T(n, k) = least unused positive value (reading rows from left to right) such that each triple of pairwise adjacent terms sums to a prime.
%C A297615 Each term may be involved in up to six sums:
%C A297615 - T(1, 1) is involved in one sum,
%C A297615 - For any r > 1, T(r, 1) and T(r, r) are involved in three sums:
%C A297615 - For any r > 1 and c such that 1 < c < r, T(r, c) is involved in six sums.
%C A297615 Among each triple of pairwise adjacent terms, we cannot have all values equal mod 3 or all values distinct mod 3; this gives rise to the patterns visible in the illustration in the Links section.
%C A297615 T(n, k) is odd iff n + k == 2 mod 3.
%C A297615 See also A297673 for a similar triangle.
%H A297615 Rémy Sigrist, <a href="/A297615/b297615.txt">Rows n = 1..100, flattened</a>
%H A297615 Rémy Sigrist, <a href="/A297615/a297615.gp.txt">PARI program for A297615</a>
%H A297615 Rémy Sigrist, <a href="/A297615/a297615.png">Colored representation of the first 500 rows</a> (where the color is function of T(n, k) mod 3)
%e A297615 Triangle begins:
%e A297615    1:                       1
%e A297615    2:                     2,  4
%e A297615    3:                   6,  5,  8
%e A297615    4:                 3, 20, 16,  7
%e A297615    5:              10, 18, 23, 14, 22
%e A297615    6:            12,  9, 26, 24, 35, 32
%e A297615    7:          11, 38, 36, 17, 30, 42, 15
%e A297615    8:        28, 34, 29, 44, 66, 31, 40, 46
%e A297615    9:      48, 21, 76, 58, 47, 54, 78, 45, 60
%e A297615   10:    13, 70, 82, 33, 88, 56, 41, 74, 62, 27
%e A297615 The term T(1, 1) = 1 is involved in the following sum:
%e A297615   -  1 +  2 +  4 =   7.
%e A297615 The term T(4, 4) = 7 is involved in the following sums:
%e A297615   -  8 + 16 +  7 =  31,
%e A297615   - 16 +  7 + 14 =  37,
%e A297615   -  7 + 14 + 22 =  43.
%e A297615 The term T(7, 6) = 42 is involved in the following sums:
%e A297615   - 35 + 32 + 42 = 109,
%e A297615   - 35 + 30 + 42 = 107,
%e A297615   - 32 + 42 + 15 =  89,
%e A297615   - 30 + 42 + 31 = 103,
%e A297615   - 42 + 31 + 40 = 113,
%e A297615   - 42 + 15 + 40 =  97.
%o A297615 (PARI) See Links section.
%Y A297615 Cf. A297673.
%K A297615 nonn,tabl
%O A297615 1,2
%A A297615 _Rémy Sigrist_, Jan 01 2018