A162876 Twin prime pairs p, p+2 such that p-1 and p+3 are both squarefree.

3, 5, 11, 13, 59, 61, 71, 73, 107, 109, 179, 181, 191, 193, 227, 229, 311, 313, 419, 421, 431, 433, 599, 601, 659, 661, 827, 829, 1019, 1021, 1031, 1033, 1091, 1093, 1319, 1321, 1427, 1429, 1487, 1489, 1607, 1609, 1619, 1621, 1787, 1789, 1871, 1873, 1931

Offset

1,1

Comments

By definition, the lower member, here at the odd-indexed positions, is in A089188.

p+1 must be divisible by 4. - Robert Israel, Jul 24 2015

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

{(p,p+2) : p in A001359, and p-1 in A005117, and p+3 in A005117}.

Example

(179,181) are in the sequence because 179-1=2*89 is squarefree and 181+1=2*7*13 is also squarefree.

Maple

f:= p -> if isprime(p) and isprime(p+2) and numtheory:-issqrfree(p-1) and numtheory:-issqrfree(p+3) then (p, p+2) else NULL fi:
map(f, [4*k-1 $ k=1..1000]); # Robert Israel, Jul 24 2015

Mathematica

f[n_]:=Module[{a=m=0}, Do[If[FactorInteger[n][[m, 2]]>1, a=1], {m, Length[FactorInteger[n]]}]; a]; lst={}; Do[p=Prime[n]; r=p+2; If[PrimeQ[r], If[f[p-1]==0&&f[r+1]==0, AppendTo[lst, p]; AppendTo[lst, r]]], {n, 7!}]; lst

Crossrefs

Cf. A089189, A089194, A162870, A162872, A162873, A162874, A162875

Sequence in context: A006794 A032457 A122564 * A162875 A166564 A058595

Adjacent sequences: A162873 A162874 A162875 * A162877 A162878 A162879

Keyword

nonn

Author

Vladimir Joseph Stephan Orlovsky, Jul 15 2009

Extensions

Definition rephrased by R. J. Mathar, Jul 27 2009

Status

approved