A173979 -id:A173979 - cp's OEIS frontend

A173980 a(n) is the smallest term m in A173978 for which A020639(2m-3) = prime(n), n > 1.

6, 19, 40, 106, 112, 265, 220, 427, 625, 730, 871, 1252, 1141, 1717, 2095, 2332, 2716, 2380, 3445, 6097, 4465, 4027, 6187, 6646, 6415, 7675, 6796, 7141, 15991, 8701, 9106, 12400, 12025, 11251, 12610, 14995, 14101, 16117, 16696, 16201, 21631, 19006, 22486, 21967

Offset: 2

Vladimir Shevelev, Mar 04 2010

nonn

Amiram Eldar, Table of n, a(n) for n = 2..1000
Vladimir Shevelev, Theorems on twin primes-dual case, arXiv:0912.4006 [math.GM], 2009-2014.

Cf. A020639, A173978, A173977, A173979.

m = 50; v = Table[0, {m}]; lpf[n_] := FactorInteger[n][[1, 1]]; aQ[n_] := (! PrimeQ[2 n - 3] || ! PrimeQ[2 n - 1]) && lpf[2 n - 3] < lpf[2 n - 1]; c = 0; n = 1; While[c < m - 1, If[aQ[n], s = PrimePi[lpf[2 n - 3]]; If[s > 1 && s <= m && v[[s]] == 0, v[[s]] = n; c++]]; n++]; Rest[v] (* Amiram Eldar, Sep 12 2019 *)

More terms from Amiram Eldar, Sep 12 2019

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

Original entry on oeis.org

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

A173980 a(n) is the smallest term m in A173978 for which A020639(2m-3) = prime(n), n > 1.

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