A162870 Primes p such that p-1 and p+1 each contain at least one cubed prime in their prime factorization.

919, 1999, 2647, 2663, 2969, 3511, 3833, 3943, 4751, 6857, 9127, 10313, 11287, 11719, 12041, 12583, 13033, 13337, 13879, 14249, 14633, 15497, 15607, 16903, 18089, 18199, 18251, 18521, 19751, 20249, 20359, 20681, 21751, 21977, 22409

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Jul 15 2009

Keywords

• nonn

Comments

The selection criterion is that p-1 and p+1 are in the subsequence 8=2^3, 24=2^3*3, 27=3^3, 40=2^3*5, 54=2*3^3,... of cubeful numbers (A046099) which actually display at least one cube in their standard prime factorization (A176297).

So at least one of the e_i in p-1=product p_i^e_i, and at least one of the e_j in p+1=product p_j^e_j must equal 3. This is more restrictive than being cubeful, so the sequence becomes a subsequence of A086708.

Examples

			271 is not in the sequence although 271 - 1 = 2*3^3*5 contains a third cube in the prime factorization, because 271 + 1 = 2^4*17 does not.
919 is in the sequence because 919 - 1 = 2*3^3*17 contains a third cube in the prime factorization and so does 919 + 1 = 2^3*5*23.
		

Links

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

Crossrefs

Cf. A089189, A089194.

Programs

Maple

isA162870 := proc(n)
    if isprime(n) then
        isA176297(n-1) and isA176297(n+1) ;
    else
        false;
    end if;
end proc:
for n from 1 to 40000 do
    if isA162870(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Dec 08 2015
N:= 10^6: # to get all terms < N, where N is even
V:= Vector(N/2):
for i from 1 do
  p:= ithprime(i);
  if p^3 > N+1 then break fi;
  if p = 2 then inds:= 4*[seq(i,i=1..floor(N/8),2)]
  else inds:= p^3*select(t -> t mod p <> 0, [$1..floor(N/2/p^3)])
  fi;
  V[inds]:= 1;
od:
select(t -> V[(t-1)/2] = 1 and V[(t+1)/2] = 1 and isprime(t), [seq(t,t=3..N,2)]); # Robert Israel, Dec 08 2015

Mathematica

f[n_]:=Module[{a=m=0},Do[If[FactorInteger[n][[m,2]]==3,a=1],{m,Length[FactorInteger[n]]}]; a]; lst={};Do[p=Prime[n];If[f[p-1]==1&&f[p+1]==1,AppendTo[lst,p]], {n,7!}];lst

Extensions

Role of cubefree numbers clarified by R. J. Mathar, Jul 31 2007