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

1,1

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.

Links

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

Example

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.

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

Crossrefs

Cf. A089189, A089194.

Sequence in context: A020399 A084843 A059668 * A251133 A083142 A332191

Adjacent sequences: A162867 A162868 A162869 * A162871 A162872 A162873

Keyword

nonn

Author

Vladimir Joseph Stephan Orlovsky, Jul 15 2009

Extensions

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

Status

approved