A173980 -id:A173980 - cp's OEIS frontend

A174453 a(n) is the smallest k >= 1 for which gcd(m + (-1)^m, m + n - 4) > 1, where m = n + k - 1.

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, 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, 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

Offset: 5

Vladimir Shevelev, Mar 20 2010

nonn

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.

Paul Tek, Table of n, a(n) for n = 5..10000
V. Shevelev, Theorems on twin primes-dual case, arXiv:0912.4006 [math.GM], 2009-2014.

Cf. A173980, A020639, A173978, A173977, A173979, A174217, A174216, A174214, A174215, A166945, A167495.

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

a[n_] := For[k=1, True, k++, m=n+k-1; If[GCD[m+(-1)^m, m+n-4]>1, Return[k]] ];
Table[a[n], {n, 5, 106}] (* Jean-François Alcover, Nov 29 2017 *)

Terms beyond a(34) from R. J. Mathar, Nov 04 2010

cp's OEIS Frontend

A174453 a(n) is the smallest k >= 1 for which gcd(m + (-1)^m, m + n - 4) > 1, where m = n + k - 1.

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