A086708 Primes p such that p-1 and p+1 are both divisible by cubes (other than 1).

271, 487, 593, 751, 809, 919, 1249, 1567, 1783, 1889, 1999, 2647, 2663, 2753, 2969, 3079, 3511, 3617, 3727, 3833, 3943, 4049, 4159, 4481, 4591, 4751, 4801, 5023, 6857, 6967, 7937, 8263, 8369, 9127, 9343, 10289, 10313, 10529, 10639, 11071, 11177

Offset: 1

Author

Jason Earls and Amarnath Murthy, Jul 28 2003

Keywords

• nonn

Links

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

Crossrefs

Cf. A162870 (subsequence).

Programs

Maple

isA086708 := proc(n)
    if isprime(n) then
        isA046099(n-1) and isA046099(n+1) ;
    else
        false;
    end if;
end proc:
n := 1:
for c from 1 to 50000 do
    if isA086708(c) then
        printf("%d %d\n",n,c) ;
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Dec 08 2015
Res:= NULL: count:= 0:
p:= 1:
while count < 100 do
  p:= nextprime(p);
  if max(seq(t[2],t=ifactors(p-1)[2]))>=3 and max(seq(t[2],t=ifactors(p+1)[2]))>=3 then
    count:= count+1; Res:= Res, p;
  fi
od:
Res; # Robert Israel, Jul 11 2018

Mathematica

f[n_]:=Max[Last/@FactorInteger[n]]; lst={};Do[p=Prime[n];If[f[p-1]>=3&&f[p+1]>=3,AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 03 2009 *)
dbcQ[p_]:=AnyTrue[Surd[#,3]&/@Rest[Divisors[p-1]],IntegerQ]&&AnyTrue[Surd[#,3]&/@Rest[ Divisors[ p+1]],IntegerQ]; Select[ Prime[Range[1500]],dbcQ] (* Harvey P. Dale, Sep 21 2024 *)

PARI

\\ Input no. of iterations n, power p and number to subtract and add k.
powerfreep4(n,p,k) = { c=0; pc=0; forprime(x=2,n, pc++; if(!ispowerfree(x-k,p) && !ispowerfree(x+k,p), c++; print1(x","); ) ); print(); print(c","pc","c/pc+.0) }
ispowerfree(m,p1) = { flag=1; y=component(factor(m),2); for(i=1,length(y), if(y[i] >= p1,flag=0;break); ); return(flag) } \\ Cino Hilliard, Dec 08 2003

Formula

{p in A000040: p+1 in A046099 and p-1 in A046099}. - R. J. Mathar, Dec 08 2015

A089199 INTERSECT A089200. - R. J. Mathar, Dec 08 2015

Extensions

Definition clarified by Harvey P. Dale, Sep 21 2024