A381322 Least prime pn such that there is a set p1 < p2 < ... < pn of primes such that, for any distinct p and q in the set, (p + q)/2 is prime.

2, 7, 19, 71, 113, 383, 509, 881, 2963, 6991, 18521, 27611

Offset

1,1

Comments

Balog shows that a(n) exists for each n. E. Bombieri showed that a(12) <= 434717 with the set {5, 17, 521, 42281, 138461, 195137, 204137, 221537, 363497, 367001, 414737, 434717}.

Links

Table of n, a(n) for n=1..12.

Antal Balog, Linear equations in primes, Mathematika 39 (1992), no. 2, pp. 367-378.

Example

Here are sets {p1, ..., pn} with pn minimal. Note that for n = 1..12, the set is unique.
  1: {2}
  2: {3, 7}
  3: {3, 7, 19}
  4: {3, 11, 23, 71}
  5: {5, 29, 53, 89, 113}
  6: {3, 11, 83, 131, 251, 383}
  7: {5, 29, 113, 269, 353, 449, 509}
  8: {5, 17, 41, 101, 257, 521, 761, 881}
  9: {23, 431, 503, 683, 863, 1091, 1523, 1871, 2963}
 10: {31, 1123, 1471, 1723, 3463, 3571, 4651, 5563, 5743, 6991}
 11: {1181, 1553, 4373, 5801, 6173, 7853, 11393, 12473, 12821, 17093, 18521}
 12: {71, 1163, 1283, 2663, 4523, 5651, 9311, 13883, 13931, 14423, 25943, 27611}

Crossrefs

Cf. A362629.

Sequence in context: A243279 A362097 A080873 * A126162 A054423 A137990

Adjacent sequences: A381319 A381320 A381321 * A381323 A381324 A381325

Keyword

nonn,hard,more

Author

Charles R Greathouse IV, Feb 25 2025

Status

approved