OFFSET
2,1
COMMENTS
a(n) < (prime(n))^3 yields an infinity of twin primes (it is sufficient, if this inequality holds for an arbitrary infinite subsequence n = n_k). For a proof, see the Shevelev link (Remark 8).
The author apparently claims to have proved the infinitude of twin primes. No alleged proof has been accepted by the mathematical community. - Jens Kruse Andersen, Jul 13 2014
In the statistical part of my link (Section 14), using the Chinese Remainder and Tolev's theorems, I reduced the supposition of the finiteness of twin primes to an arbitrarily long coin-flipping experiment in which only "heads" appear. There I gave only a "demonstration" of the infinity of twin primes. In the analytical part (Sections 15-18) I proved unconditionally till now only Theorem 13. - Vladimir Shevelev, Jul 22 2014
LINKS
Jens Kruse Andersen, Table of n, a(n) for n = 2..10000
V. Shevelev, Theorems on twin primes-dual case, arXiv:0912.4006 [math.GM], 2010-2014. (Sections 10,14-18). [Note this article has been changed many times.]
MATHEMATICA
a[n_, k_] := For[p = Prime[n], True, p = NextPrime[p], If[PrimeQ[p Prime[n] + k], Return[p]]];
a[n_] := Max[a[n, -2], a[n, 2]];
Table[a[n], {n, 2, 60}] (* Jean-François Alcover, Nov 18 2018 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Shevelev, Jun 30 2014
EXTENSIONS
More terms from Peter J. C. Moses, Jun 30 2014
STATUS
approved