A297673 -id:A297673 - cp's OEIS frontend

A296305 Triangular array T(n, k) read by rows, n > 0, 0 < k <= n: T(n, k) = A297673(n, k) + A297673(n+1, k) + A297673(n+1, k+1).

7, 11, 17, 17, 23, 29, 31, 37, 31, 43, 43, 53, 53, 61, 59, 53, 71, 79, 79, 71, 71, 71, 79, 89, 89, 89, 97, 107, 97, 107, 127, 127, 109, 113, 127, 127, 127, 137, 137, 163, 149, 131, 139, 157, 163, 149, 167, 163, 173, 173, 157, 163, 179, 193, 197, 173, 211, 211

Offset: 1

Rémy Sigrist, Jan 03 2018

All terms are odd prime numbers.

			Triangle begins:
   1:                                 7
   2:                             11,   17
   3:                          17,   23,   29
   4:                       31,   37,   31,   43
   5:                    43,   53,   53,   61,   59
   6:                 53,   71,   79,   79,   71,   71
   7:              71,   79,   89,   89,   89,   97,  107
   8:           97,  107,  127,  127,  109,  113,  127,  127
   9:       127,  137,  137,  163,  149,  131,  139,  157,  163
  10:    149,  167,  163,  173,  173,  157,  163,  179,  193,  197

Rémy Sigrist, Rows n = 1..100, flattened
Rémy Sigrist, PARI program for A296305

Cf. A297673.

PARI
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See Links section.
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A297615 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 each triple of pairwise adjacent terms sums to a prime.

Original entry on oeis.org

1, 2, 4, 6, 5, 8, 3, 20, 16, 7, 10, 18, 23, 14, 22, 12, 9, 26, 24, 35, 32, 11, 38, 36, 17, 30, 42, 15, 28, 34, 29, 44, 66, 31, 40, 46, 48, 21, 76, 58, 47, 54, 78, 45, 60, 13, 70, 82, 33, 88, 56, 41, 74, 62, 27, 50, 68, 59, 52, 72, 19, 136, 64, 43, 92, 80, 84

Offset: 1

Rémy Sigrist, Jan 01 2018

Each term may be involved in up to six sums:
- T(1, 1) is involved in one sum,
- For any r > 1, T(r, 1) and T(r, r) are involved in three sums:
- For any r > 1 and c such that 1 < c < r, T(r, c) is involved in six sums.
Among each triple of pairwise adjacent terms, we cannot have all values equal mod 3 or all values distinct mod 3; this gives rise to the patterns visible in the illustration in the Links section.
T(n, k) is odd iff n + k == 2 mod 3.
See also A297673 for a similar triangle.

			Triangle begins:
   1:                       1
   2:                     2,  4
   3:                   6,  5,  8
   4:                 3, 20, 16,  7
   5:              10, 18, 23, 14, 22
   6:            12,  9, 26, 24, 35, 32
   7:          11, 38, 36, 17, 30, 42, 15
   8:        28, 34, 29, 44, 66, 31, 40, 46
   9:      48, 21, 76, 58, 47, 54, 78, 45, 60
  10:    13, 70, 82, 33, 88, 56, 41, 74, 62, 27
The term T(1, 1) = 1 is involved in the following sum:
  -  1 +  2 +  4 =   7.
The term T(4, 4) = 7 is involved in the following sums:
  -  8 + 16 +  7 =  31,
  - 16 +  7 + 14 =  37,
  -  7 + 14 + 22 =  43.
The term T(7, 6) = 42 is involved in the following sums:
  - 35 + 32 + 42 = 109,
  - 35 + 30 + 42 = 107,
  - 32 + 42 + 15 =  89,
  - 30 + 42 + 31 = 103,
  - 42 + 31 + 40 = 113,
  - 42 + 15 + 40 =  97.

Rémy Sigrist, Rows n = 1..100, flattened
Rémy Sigrist, PARI program for A297615
Rémy Sigrist, Colored representation of the first 500 rows (where the color is function of T(n, k) mod 3)

Cf. A297673.

PARI
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See Links section.
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cp's OEIS Frontend

A296305 Triangular array T(n, k) read by rows, n > 0, 0 < k <= n: T(n, k) = A297673(n, k) + A297673(n+1, k) + A297673(n+1, k+1).

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