A089194 Primes p such that p-1 and p+1 are cube- or higher power-free.

2, 3, 5, 11, 13, 19, 29, 37, 43, 59, 61, 67, 83, 101, 131, 139, 149, 157, 173, 179, 181, 197, 211, 227, 229, 277, 283, 293, 307, 317, 331, 347, 349, 373, 389, 397, 419, 421, 443, 461, 467, 491, 509, 523, 547, 557, 563, 571, 587, 613, 619, 643, 653, 659, 661

Offset: 1

Author

Cino Hilliard, Dec 08 2003

Keywords

• easy
• nonn

Comments

A212793(a(n) - 1) = A212793(a(n) + 1) = 1. - Reinhard Zumkeller, May 27 2012

Examples

			43 is included because 43 - 1 = 2 * 3 * 7 and 43 + 1 = 2^2 * 11 are both cubefree.
71 is omitted because the p+1 side, 72 = 2^3 * 3^2, has a cube factor.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A004709. Subsequence of A089189.

Programs

Haskell

a089194 n = a089194_list !! (n-1)
a089194_list = filter ((== 1) . a212793 . (+ 1)) a097375_list
-- Reinhard Zumkeller, May 27 2012

Maple

isA089194 := proc(n)
    if isprime(n) then
        isA004709(n-1) and isA004709(n+1) ;
    else
        false;
    end if;
end proc: # R. J. Mathar, 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]==0&&f[p+1]==0,AppendTo[lst,p]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 15 2009 *)
p3fQ[n_]:=Max[Transpose[FactorInteger[n]][[2]]]<3; Select[Prime[Range[ 200]], AllTrue[#+{1,-1},p3fQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 08 2015 *)

PARI

\\ input number of iterations n, power p and the number to subtract k.
powerfreep2(n,p,d) = { c=0; pc=0; forprime(x=2,n, pc++; if(ispowerfree(x-d,p) && ispowerfree(x+d,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) }

Formula

{p in A000040: p+1 in A004709 and p-1 in A004709}. - R. J. Mathar, Dec 08 2015