A073671 -id:A073671 - cp's OEIS frontend

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.

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

Offset: 1

Rémy Sigrist, Jan 03 2018

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.

			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.

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

Cf. A073671, A076990, A297615, A296305.

PARI
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See Links section.
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cp's OEIS Frontend

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.

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