A162872 Primes p such that p-1 and p+1 each contain at least one squared prime in their prime factorization.

19, 149, 197, 199, 293, 307, 349, 491, 523, 557, 577, 739, 773, 883, 1013, 1051, 1061, 1151, 1171, 1277, 1451, 1493, 1531, 1549, 1601, 1637, 1667, 1693, 1709, 1733, 1747, 1861, 1949, 2069, 2141, 2179, 2251, 2351, 2357, 2467, 2549, 2683, 2789, 2843, 2851

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 4=2^2, 9=3^2, 12=2^2*3, 18=2*3^2, ... of nonsquarefree numbers (A013929) that actually display at least one square in their standard prime factorization.

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 2. This is more stringent than being nonsquarefree, and the sequence becomes a subsequence of A075432.

Examples

			19 is in the sequence because 19 - 1 = 2*3^2 contains 3^2 and because 19 + 1 = 2^2*5 contains 2^2 in the factorization.
		

Links

• R. J. Mathar and Robert Israel, Table of n, a(n) for n = 1..10000 (n = 1..536 from R. J. Mathar)

Crossrefs

Cf. A038109, A162870.

Programs

Maple

isA162872 := proc(n)
    if isprime(n) then
        isA038109(n-1) and isA038109(n+1) ;
    else
        false;
    end if;
end proc:
n := 1:
for c from 1 to 50000 do
    if isA162872(c) then
        printf("%d %d\n",n,c) ;
        n := n+1 ;
end if; # R. J. Mathar, Dec 08 2015
N:= 10^5: # to get all terms < N, where N is even
V:= Vector(N/2):
for i from 1 do
  p:= ithprime(i);
  if p^2 > N+1 then break fi;
  if p = 2 then inds:= 2*[seq(i, i=1..floor(N/4), 2)]
  else inds:= p^2*select(t -> t mod p <> 0, [$1..floor(N/2/p^2)])
  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]]==2,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
ospQ[n_]:=AnyTrue[FactorInteger[n+1][[;;,2]],#==2&]&&AnyTrue[FactorInteger[n-1][[;;,2]],#==2&]; Select[Prime[Range[500]],ospQ] (* Harvey P. Dale, May 11 2025 *)

Formula

{p in A000040: p+1 in A038109 and p-1 in A038109}. - R. J. Mathar, Dec 08 2015

Extensions

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