%I #19 Jan 04 2018 09:15:56

%S 1,2,4,3,6,7,5,9,8,14,10,16,12,11,18,13,20,17,24,26,15,19,21,30,32,23,

%T 22,34,25,27,31,28,29,37,38,35,33,39,41,55,44,36,40,49,43,42,52,46,50,

%U 58,47,48,51,57,63,45,62,53,64,59,56,54,61,67,69,65,60

%N 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 T(n, k) + T(n+1, k) + T(n+1, k+1) is prime.

%C See A296305 for the corresponding sums.

%C Each term may be involved in up to three sums:

%C - T(1, 1) is involved in one sum,

%C - For any n > 1, T(n, 1) and T(n, k) are involved in two sums:

%C - For any n > 1 and k such that 1 < k < n, T(n, k) is involved in three sums.

%C The parity of the terms of the triangle has interesting features:

%C - For any n > 35:

%C - T(n, 1) is even,

%C - T(n, k) is odd for any k such that 1 < k < n - 34,

%C - T(n, n - 34) is even,

%C - T(n, n - k) and T(n + 64, n + 64 - k) have the same parity for k=0..34,

%C - See representation in Links section (the black pattern visible alongside the right border is eventually periodic),

%C - These features also appear in the scatterplot of the triangle as a flat sequence in the form of two branches: the first branch above the X=Y axis corresponds to the (frequent) odd terms, and the dashed branch under the X=Y axis corresponds to the (sparse) even terms.

%C This triangle has building features in common with A073671 and with A076990:

%C - for three distinct positive numbers to sum to a prime number, either all of them are odd or two of them are even and the other is odd,

%C - we have both situations here,

%C - we have only the first situation in A073671,

%C - we have only the second situation in A076990.

%C See also A297615 for a similar triangle.

%H Rémy Sigrist, <a href="/A297673/b297673.txt">Rows n = 1..100, flattened</a>

%H Rémy Sigrist, <a href="/A297673/a297673.png">Colored representation of the first 500 rows</a> (where the color is function of the parity of T(n, k))

%H Rémy Sigrist, <a href="/A297673/a297673.gp.txt">PARI program for A297673</a>

%e Triangle begins:

%e 1: 1

%e 2: 2, 4

%e 3: 3, 6, 7

%e 4: 5, 9, 8, 14

%e 5: 10, 16, 12, 11, 18

%e 6: 13, 20, 17, 24, 26, 15

%e 7: 19, 21, 30, 32, 23, 22, 34

%e 8: 25, 27, 31, 28, 29, 37, 38, 35

%e 9: 33, 39, 41, 55, 44, 36, 40, 49, 43

%e 10: 42, 52, 46, 50, 58, 47, 48, 51, 57, 63

%e The term T(1, 1) = 1 is involved in the following sum:

%e - 1 + 2 + 4 = 7.

%e The term T(3, 3) = 7 is involved in the following sums:

%e - 4 + 6 + 7 = 17,

%e - 7 + 8 + 14 = 29.

%e The term T(4, 2) = 9 is involved in the following sums:

%e - 3 + 5 + 9 = 17,

%e - 6 + 9 + 8 = 23,

%e - 9 + 16 + 12 = 37.

%o (PARI) See Links section.

%Y Cf. A073671, A076990, A297615, A296305.

%K nonn,tabl

%O 1,2

%A _Rémy Sigrist_, Jan 03 2018