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

%I A086708 #37 Sep 21 2024 17:58:43
%S A086708 271,487,593,751,809,919,1249,1567,1783,1889,1999,2647,2663,2753,2969,
%T A086708 3079,3511,3617,3727,3833,3943,4049,4159,4481,4591,4751,4801,5023,
%U A086708 6857,6967,7937,8263,8369,9127,9343,10289,10313,10529,10639,11071,11177
%N A086708 Primes p such that p-1 and p+1 are both divisible by cubes (other than 1).
%H A086708 Robert Israel, <a href="/A086708/b086708.txt">Table of n, a(n) for n = 1..10000</a>
%F A086708 {p in A000040: p+1 in A046099 and p-1 in A046099}. - _R. J. Mathar_, Dec 08 2015
%F A086708 A089199 INTERSECT A089200. - _R. J. Mathar_, Dec 08 2015
%p A086708 isA086708 := proc(n)
%p A086708     if isprime(n) then
%p A086708         isA046099(n-1) and isA046099(n+1) ;
%p A086708     else
%p A086708         false;
%p A086708     end if;
%p A086708 end proc:
%p A086708 n := 1:
%p A086708 for c from 1 to 50000 do
%p A086708     if isA086708(c) then
%p A086708         printf("%d %d\n",n,c) ;
%p A086708         n := n+1 ;
%p A086708     end if;
%p A086708 end do: # _R. J. Mathar_, Dec 08 2015
%p A086708 Res:= NULL: count:= 0:
%p A086708 p:= 1:
%p A086708 while count < 100 do
%p A086708   p:= nextprime(p);
%p A086708   if max(seq(t[2],t=ifactors(p-1)[2]))>=3 and max(seq(t[2],t=ifactors(p+1)[2]))>=3 then
%p A086708     count:= count+1; Res:= Res, p;
%p A086708   fi
%p A086708 od:
%p A086708 Res; # _Robert Israel_, Jul 11 2018
%t A086708 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 *)
%t A086708 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 *)
%o A086708 (PARI)
%o A086708 \\ Input no. of iterations n, power p and number to subtract and add k.
%o A086708 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) }
%o A086708 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
%Y A086708 Cf. A162870 (subsequence).
%K A086708 nonn
%O A086708 1,1
%A A086708 _Jason Earls_ and _Amarnath Murthy_, Jul 28 2003
%E A086708 Definition clarified by _Harvey P. Dale_, Sep 21 2024