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

Author

Vladimir Joseph Stephan Orlovsky, Jul 15 2009

Keywords

• nonn

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

Examples

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

Links

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

Crossrefs

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

Programs

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

Formula

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

Extensions

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