A307782 Even integers k such that there exists a prime p with p=min{q: q prime and (k-q) prime} and k < p^3.

4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 36, 38, 42, 48, 52, 54, 58, 60, 66, 68, 72, 78, 80, 84, 88, 90, 94, 96, 98, 102, 108, 114, 118, 120, 122, 124, 126, 128, 138, 146, 148, 150, 158, 164, 174, 180, 188, 190, 192, 206, 208, 210, 212, 218, 220, 222, 224, 240, 248, 250, 252, 258, 264, 270, 278, 290, 292, 294, 300, 302, 304, 306, 308, 324, 326, 328, 330, 332, 338, 346

Offset: 1

Author

Corinna Regina Böger, Apr 28 2019

Keywords

• nonn

Comments

[Please keep the larger data section as it shows where the sequence first differs from A093161.]

This is another member of a family of sequences concerning the Strong Goldbach Conjecture, which I define as follows: Let (x, y, z) be real numbers with x >= 2, y > 0, z >= 0. An even integer k is then called an (x, y, z) Extraordinary Goldbach Number (EGN) if there exists a prime p with p=min{q: q prime and (k-q) prime} and (k - z*p) < y*p^x. a(n) represents the (3, 1, 0) extraordinary Goldbach numbers. A093161 consists of (3, 1, 1) EGN, A307542 are the (2, 1, 1) EGN, A279040 are the (2, 2, 0) EGN and A244408 are the (2, 1, 0).

a(104809) is very probably the last term and there are no more terms below 4*10^18.

There are only 11 terms in A093161 that are not in this sequence; these are 344, 1338, 12184, 12186, 24400, 148912, 1030342, 2571406, 3308008, 5929868, 15813352.

Examples

			344 is not in the sequence, because the smallest prime p for 344 is 7 with 7^3 = 343 < 344, whereas it is in A093161 due to 344 - 7 = 337 < 7^3.
		

Links

• Corinna Regina Böger, Table of n, a(n) for n = 1..10000
• Corinna Regina Böger, a-file, Table of n, a(n) for n = 1..104809

Crossrefs

Cf. A093161, A244408, A279040, A307542.

Programs

PARI

extraordinaryGoldbach(x,y,z,k) = forprime(p=2, k/2, if(isprime(k-p), if(y*p^x+z*p>=k, return(1),return(0)))); 0
is(n) = n%2 == 0 && extraordinaryGoldbach(3, 1, 0, n)