A174215 -id:A174215 - cp's OEIS frontend

A174216 a(1)=15; for n>1, a(n) = the smallest number k >a(n-1) such that 2A174214(k)= 3(k-1).

15, 27, 63, 123, 279, 567, 1143, 2307, 4623, 9447, 18927, 38283, 77139, 154839, 309747, 620463, 1241823, 2483847, 4967739, 9935607, 19892547, 39785199

Offset: 1

Vladimir Shevelev, Mar 12 2010

Theorem: If the sequence is infinite, then there exist infinitely many twin primes.

Conjecture. a(n+1)/a(n) tends to 2.

V. Shevelev, Theorems on twin primes-dual case, arXiv:0912.4006 [math.GM], 2009-2014.

Cf. A174214, A174215, A166945, A167495.

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

(* 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 *)

Terms from a(11) on corrected by R. J. Mathar, Mar 16 2010
I corrected the terms beginning with a(11) and added some new terms. - Vladimir Shevelev, Mar 27 2010
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

A174217 a(n) = (A174216(n)-1)/2.

Original entry on oeis.org

7, 13, 31, 61, 139, 283, 571, 1153, 2311, 4723, 9463, 19141, 38569, 77419, 154873, 310231, 620911, 1241923, 2483869, 4967803, 9946273, 19892599

Offset: 1

Vladimir Shevelev, Mar 12 2010

Related to the generation of twin primes according to section 6 of the preprint.

V. Shevelev, Theorems on twin primes-dual case, arXiv:0912.4006 [math.GM], 2009-2014.

Cf. A174214, A174215, A174216, A166945, A167495.

(* b = A174214 *) b[n_] := b[n] = Which[n == 9, 14, CoprimeQ[b[n - 1], n - 1 - (-1)^n], b[n - 1] + 1, True, 2 n - 4];
(* c = A174216 *) c[n_] := c[n] = If[n == 1, 15, For[k = c[n - 1] + 1, True, k++, If[2 b[k] == 3 (k - 1), Return[k]]]];
Table[a[n] = (c[n] - 1)/2; Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 22}] (* Jean-François Alcover, Jan 29 2019 *)

A174214(A174216(n)) = 3*a(n), n>1.

Terms after a(11) corrected by Vladimir Shevelev, Nov 02 2010

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

A174216 a(1)=15; for n>1, a(n) = the smallest number k >a(n-1) such that 2A174214(k)= 3(k-1).

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A174217 a(n) = (A174216(n)-1)/2.

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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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cp's OEIS Frontend

A174216 a(1)=15; for n>1, a(n) = the smallest number k >a(n-1) such that 2*A174214(k)= 3*(k-1).

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A174217 a(n) = (A174216(n)-1)/2.

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Author

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Mathematica

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Extensions

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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Author

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A174216 a(1)=15; for n>1, a(n) = the smallest number k >a(n-1) such that 2A174214(k)= 3(k-1).