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%I #51 Nov 18 2018 09:23:10
%S 3,7,11,23,17,37,23,41,43,61,47,61,53,73,109,107,89,73,109,227,113,
%T 113,139,157,127,149,127,131,283,137,139,181,173,179,167,191,181,227,
%U 193,251,239,199,233,257,239,251,239,241,271,313,241,271,281,277,443,389
%N a(n) = max(A244570(n), A244571(n)).
%C 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).
%C 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
%C 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
%H Jens Kruse Andersen, <a href="/A244572/b244572.txt">Table of n, a(n) for n = 2..10000</a>
%H V. Shevelev, <a href="https://arxiv.org/abs/0912.4006">Theorems on twin primes-dual case</a>, arXiv:0912.4006 [math.GM], 2010-2014. (Sections 10,14-18). [Note this article has been changed many times.]
%t a[n_, k_] := For[p = Prime[n], True, p = NextPrime[p], If[PrimeQ[p Prime[n] + k], Return[p]]];
%t a[n_] := Max[a[n, -2], a[n, 2]];
%t Table[a[n], {n, 2, 60}] (* _Jean-François Alcover_, Nov 18 2018 *)
%Y Cf. A244570, A244571, A242519, A242520.
%K nonn
%O 2,1
%A _Vladimir Shevelev_, Jun 30 2014
%E More terms from _Peter J. C. Moses_, Jun 30 2014
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