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%I #14 Dec 10 2018 08:53:15
%S 15,27,63,123,279,567,1143,2307,4623,9447,18927,38283,77139,154839,
%T 309747,620463,1241823,2483847,4967739,9935607,19892547,39785199
%N a(1)=15; for n>1, a(n) = the smallest number k >a(n-1) such that 2*A174214(k)= 3*(k-1).
%C Theorem: If the sequence is infinite, then there exist infinitely many twin primes.
%C Conjecture. a(n+1)/a(n) tends to 2.
%H V. Shevelev, <a href="http://arXiv.org/abs/0912.4006">Theorems on twin primes-dual case</a>, arXiv:0912.4006 [math.GM], 2009-2014.
%p A174216 := proc(n) option remember ; if n =1 then 15 ; else for k from procname(n-1)+1 do if 2*A173214(k) = 3*(k-1) then return k; end if; end do ; end if; end proc: # _R. J. Mathar_, Mar 16 2010
%t (* b = A174214 *) b[n_] := b[n] = Which[n==9, 14, CoprimeQ[b[n-1], n-1- (-1)^n], b[n-1]+1, True, 2n-4]; a[n_] := a[n] = If[n==1, 15, For[k = a[n- 1]+1, True, k++, If[2b[k] == 3(k-1), Return[k]]]]; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 22}] (* _Jean-François Alcover_, Feb 02 2016 *)
%Y Cf. A174214, A174215, A166945, A167495.
%K nonn,more
%O 1,1
%A _Vladimir Shevelev_, Mar 12 2010
%E Terms from a(11) on corrected by _R. J. Mathar_, Mar 16 2010
%E I corrected the terms beginning with a(11) and added some new terms. - _Vladimir Shevelev_, Mar 27 2010
%E Terms from a(11) onwards were corrected according to independent calculations by R. Mathar, M. Alekseyev, M. Hasler and A. Heinz (SeqFan lists 30 Oct and 1 Nov 2010). - _Vladimir Shevelev_, Nov 02 2010
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