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A297673 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.

%I A297673 #19 Jan 04 2018 09:15:56
%S A297673 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 A297673 22,34,25,27,31,28,29,37,38,35,33,39,41,55,44,36,40,49,43,42,52,46,50,
%U A297673 58,47,48,51,57,63,45,62,53,64,59,56,54,61,67,69,65,60
%N A297673 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 A297673 See A296305 for the corresponding sums.
%C A297673 Each term may be involved in up to three sums:
%C A297673 - T(1, 1) is involved in one sum,
%C A297673 - For any n > 1, T(n, 1) and T(n, k) are involved in two sums:
%C A297673 - For any n > 1 and k such that 1 < k < n, T(n, k) is involved in three sums.
%C A297673 The parity of the terms of the triangle has interesting features:
%C A297673 - For any n > 35:
%C A297673 - T(n, 1) is even,
%C A297673 - T(n, k) is odd for any k such that 1 < k < n - 34,
%C A297673 - T(n, n - 34) is even,
%C A297673 - T(n, n - k) and T(n + 64, n + 64 - k) have the same parity for k=0..34,
%C A297673 - See representation in Links section (the black pattern visible alongside the right border is eventually periodic),
%C A297673 - 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 A297673 This triangle has building features in common with A073671 and with A076990:
%C A297673 - 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 A297673 - we have both situations here,
%C A297673 - we have only the first situation in A073671,
%C A297673 - we have only the second situation in A076990.
%C A297673 See also A297615 for a similar triangle.
%H A297673 Rémy Sigrist, <a href="/A297673/b297673.txt">Rows n = 1..100, flattened</a>
%H A297673 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 A297673 Rémy Sigrist, <a href="/A297673/a297673.gp.txt">PARI program for A297673</a>
%e A297673 Triangle begins:
%e A297673    1:                       1
%e A297673    2:                     2,  4
%e A297673    3:                   3,  6,  7
%e A297673    4:                 5,  9,  8, 14
%e A297673    5:              10, 16, 12, 11, 18
%e A297673    6:            13, 20, 17, 24, 26, 15
%e A297673    7:          19, 21, 30, 32, 23, 22, 34
%e A297673    8:        25, 27, 31, 28, 29, 37, 38, 35
%e A297673    9:      33, 39, 41, 55, 44, 36, 40, 49, 43
%e A297673   10:    42, 52, 46, 50, 58, 47, 48, 51, 57, 63
%e A297673 The term T(1, 1) = 1 is involved in the following sum:
%e A297673 - 1 + 2 + 4 = 7.
%e A297673 The term T(3, 3) = 7 is involved in the following sums:
%e A297673 - 4 + 6 + 7 = 17,
%e A297673 - 7 + 8 + 14 = 29.
%e A297673 The term T(4, 2) = 9 is involved in the following sums:
%e A297673 - 3 + 5 + 9 = 17,
%e A297673 - 6 + 9 + 8 = 23,
%e A297673 - 9 + 16 + 12 = 37.
%o A297673 (PARI) See Links section.
%Y A297673 Cf. A073671, A076990, A297615, A296305.
%K A297673 nonn,tabl
%O A297673 1,2
%A A297673 _Rémy Sigrist_, Jan 03 2018