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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. 3
 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, 22, 34, 25, 27, 31, 28, 29, 37, 38, 35, 33, 39, 41, 55, 44, 36, 40, 49, 43, 42, 52, 46, 50, 58, 47, 48, 51, 57, 63, 45, 62, 53, 64, 59, 56, 54, 61, 67, 69, 65, 60 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS See A296305 for the corresponding sums. Each term may be involved in up to three sums: - T(1, 1) is involved in one sum, - For any n > 1, T(n, 1) and T(n, k) are involved in two sums: - For any n > 1 and k such that 1 < k < n, T(n, k) is involved in three sums. The parity of the terms of the triangle has interesting features: - For any n > 35: - T(n, 1) is even, - T(n, k) is odd for any k such that 1 < k < n - 34, - T(n, n - 34) is even, - T(n, n - k) and T(n + 64, n + 64 - k) have the same parity for k=0..34, - See representation in Links section (the black pattern visible alongside the right border is eventually periodic), - 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. This triangle has building features in common with A073671 and with A076990: - 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, - we have both situations here, - we have only the first situation in A073671, - we have only the second situation in A076990. See also A297615 for a similar triangle. LINKS Rémy Sigrist, Rows n = 1..100, flattened Rémy Sigrist, Colored representation of the first 500 rows (where the color is function of the parity of T(n, k)) Rémy Sigrist, PARI program for A297673 EXAMPLE Triangle begins:    1:                       1    2:                     2,  4    3:                   3,  6,  7    4:                 5,  9,  8, 14    5:              10, 16, 12, 11, 18    6:            13, 20, 17, 24, 26, 15    7:          19, 21, 30, 32, 23, 22, 34    8:        25, 27, 31, 28, 29, 37, 38, 35    9:      33, 39, 41, 55, 44, 36, 40, 49, 43   10:    42, 52, 46, 50, 58, 47, 48, 51, 57, 63 The term T(1, 1) = 1 is involved in the following sum: - 1 + 2 + 4 = 7. The term T(3, 3) = 7 is involved in the following sums: - 4 + 6 + 7 = 17, - 7 + 8 + 14 = 29. The term T(4, 2) = 9 is involved in the following sums: - 3 + 5 + 9 = 17, - 6 + 9 + 8 = 23, - 9 + 16 + 12 = 37. PROG (PARI) See Links section. CROSSREFS Cf. A073671, A076990, A297615, A296305. Sequence in context: A226246 A216623 A297551 * A083050 A194030 A083044 Adjacent sequences:  A297670 A297671 A297672 * A297674 A297675 A297676 KEYWORD nonn,tabl AUTHOR Rémy Sigrist, Jan 03 2018 STATUS approved

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Last modified June 18 01:16 EDT 2021. Contains 345098 sequences. (Running on oeis4.)