A244572 a(n) = max(A244570(n), A244571(n)).

3, 7, 11, 23, 17, 37, 23, 41, 43, 61, 47, 61, 53, 73, 109, 107, 89, 73, 109, 227, 113, 113, 139, 157, 127, 149, 127, 131, 283, 137, 139, 181, 173, 179, 167, 191, 181, 227, 193, 251, 239, 199, 233, 257, 239, 251, 239, 241, 271, 313, 241, 271, 281, 277, 443, 389

Offset: 2

Author

Vladimir Shevelev, Jun 30 2014

Keywords

• nonn

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.]

Crossrefs

Cf. A244570, A244571, A242519, A242520.

Programs

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 *)

Extensions

More terms from Peter J. C. Moses, Jun 30 2014