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%I #19 Sep 04 2018 03:36:36
%S 1,2,1,1,1,5,1,2,1,1,1,12,1,2,1,1,1,18,1,2,1,1,1,4,1,2,1,1,1,30,1,2,1,
%T 1,1,5,1,2,1,1,1,42,1,2,1,1,1,6,1,2,1,1,1,4,1,2,1,1,1,60,1,2,1,1,1,5,
%U 1,2,1,1,1,72,1,2,1,1,1,9,1,2,1,1,1,4,1,2,1,1,1,6,1,2,1,1,1,5,1,2,1,1,1,102
%N a(n) is the smallest k >= 1 for which gcd(m + (-1)^m, m + n - 4) > 1, where m = n + k - 1.
%C If a(n) > sqrt(n), then n-3 is the larger of twin primes. In these cases we have a(10)=5 and, for n > 10, a(n) = n-4. For odd n and for n == 2 (mod 6), a(n)=1; for n == 0 (mod 6), a(n)=2; for {n == 4 (mod 6)} & {n == 8 (mod 10)}, a(n)=4, etc. The problem is to develop this sieve for the excluding n for which a(n) <= sqrt(n) and to obtain nontrivial lower estimates for the counting function of the larger of twin primes.
%H Paul Tek, <a href="/A174453/b174453.txt">Table of n, a(n) for n = 5..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], 2009-2014.
%p A174453 := proc(n) local k,m ; for k from 1 do m := n+k-1 ; if igcd(m+(-1)^m,m+n-4) > 1 then return k; end if; end do: end proc: seq(A174453(n),n=5..120); # _R. J. Mathar_, Nov 04 2010
%t a[n_] := For[k=1, True, k++, m=n+k-1; If[GCD[m+(-1)^m, m+n-4]>1, Return[k]] ];
%t Table[a[n], {n, 5, 106}] (* _Jean-François Alcover_, Nov 29 2017 *)
%Y Cf. A173980, A020639, A173978, A173977, A173979, A174217, A174216, A174214, A174215, A166945, A167495.
%K nonn,uned
%O 5,2
%A _Vladimir Shevelev_, Mar 20 2010
%E Terms beyond a(34) from _R. J. Mathar_, Nov 04 2010
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