A308040 Numbers k such that k - prevprime(k-1) is prime where prevprime(n) is the largest prime < n.

4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 24, 25, 26, 28, 30, 31, 32, 33, 34, 36, 38, 39, 40, 42, 43, 44, 45, 46, 48, 49, 50, 52, 54, 55, 56, 58, 60, 61, 62, 63, 64, 66, 68, 69, 70, 72, 73, 74, 75, 76, 78, 80, 81, 82, 84, 85, 86, 88, 90, 91, 92, 94, 96, 99, 100

Offset: 1

Author

Wesley Ivan Hurt, May 10 2019

Keywords

• nonn

Comments

Contains all odd numbers k >= 5 such that k - 2 is prime.

If Goldbach's conjecture is true, the sequence contains an even number k iff there exists a Goldbach partition of k that includes the largest prime < k - 1. This sequence agrees with A014091 (numbers that are the sum of two primes) up to k = 96, but does not include 98 since the largest prime strictly below 97 is 89, which is paired with 9 (and thus, not a Goldbach partition).

Links

• Eric Weisstein's World of Mathematics, Goldbach Partition
• Wikipedia, Goldbach's conjecture
• Index entries for sequences related to Goldbach conjecture
• Index entries for sequences related to partitions

Crossrefs

Cf. A014091.

Programs

Mathematica

Select[Range[4, 100], PrimeQ[# - NextPrime[# - 1, -1]] &]