A229993 Numbers for which c(n) - 1 and c(n) + 1 are twin primes, where c(n) = A061214(n) = product of composite numbers between prime(n) and prime(n+1) .

2, 3, 5, 6, 7, 8, 10, 13, 14, 17, 20, 26, 28, 29, 33, 35, 41, 43, 45, 49, 52, 57, 60, 64, 69, 81, 83, 85, 89, 90, 91, 98, 104, 109, 113, 116, 120, 134, 140, 142, 144, 148, 152, 171, 173, 176, 178, 182, 190, 201, 202, 204, 206, 209, 212, 215, 225, 230, 234

Offset

2,1

Links

Harvey P. Dale, Table of n, a(n) for n = 2..1000

Example

c(n) - 1:  3, 5, 719, 11, 3359, 17, 9239.
c(n) + 1: 5, 7, 721, 13, 3361, 19, 9241.  Here, for example, for we have twin primes except for n = 4, since 721 is not prime.

Mathematica

z = 400; c[n_] := Product[k, {k, Prime[n] + 1, Prime[n + 1] - 1}]; d[n_] := If[PrimeQ[c[n] - 1], 1, 0]; t1 = Table[d[n], {n, 1, z}]; u1 = Flatten[Position[t1, 1]]; e[n_] := If[PrimeQ[c[n] + 1], 1, 0]; t2 = Table[e[n], {n, 1, z}]; u2 = Flatten[Position[t2, 1]]; u = Intersection[u1, u2]
pcnQ[n_]:=Module[{p=Times@@Range[Prime[n]+1, Prime[n+1]-1]}, AllTrue[p+{1, -1}, PrimeQ]]; Select[Range[250], pcnQ] (* Harvey P. Dale, Jan 28 2023 *)

Crossrefs

Cf. A061214, A229991, A229992.

Sequence in context: A028783 A274779 A120486 * A323252 A219255 A025501

Adjacent sequences: A229990 A229991 A229992 * A229994 A229995 A229996

Keyword

nonn

Author

Clark Kimberling, Oct 09 2013

Status

approved