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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.)