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

5,2

Comments

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.

Links

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.

Maple

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

Mathematica

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 *)

Crossrefs

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

Sequence in context: A014651 A275422 A169951 * A361781 A082063 A260148

Adjacent sequences: A174450 A174451 A174452 * A174454 A174455 A174456

Keyword

nonn

Author

Vladimir Shevelev, Mar 20 2010

Extensions

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

Status

approved