A257116 Smallest prime p such that none of p + 1, p + 3,... p + 2n - 1 are squarefree and all of p + 2, p + 4,... p + 2n are squarefree.

3, 17, 487, 947, 947, 38639, 38639

Offset: 1

Author

Juri-Stepan Gerasimov, Apr 25 2015

Keywords

• nonn
• fini
• full

Comments

a(8) and higher do not exist because at least one of p+2, p+4, ..., p+16 is divisible by 9 unless p is divisible by 9, in which case it is not prime. - Charles R Greathouse IV, Apr 27 2015

Examples

			a(1) = 3 because 3 + 1 = 4 is not squarefree, 3 + 2 = 5 is squarefree, 3 is prime.
		

Crossrefs

Cf. A257108, A257115.

Programs

Maple

p:= 0:
for i from 1 to 5000 do
  p:= nextprime(p);
  for n from 1 while numtheory:-issqrfree(p+2*n)
       and not numtheory:-issqrfree(p+2*n-1) do
        if not assigned(A[n]) then A[n]:= p
          fi
    od:
od:
seq(A[i],i=1..7); # Robert Israel, Apr 27 2015

Mathematica

a[n_] := For[k=1, True, k++, p = Prime[k]; r = p + Range[1, 2*n-1, 2]; If[(And @@ ((!SquareFreeQ[#])& /@ r)) && And @@ (SquareFreeQ /@ (r+1)), Return[p]]]; Table[ a[n], {n, 1, 7}] (* Jean-François Alcover, Apr 28 2015 *)

PARI

has(p,n)=for(i=1,2*n,if(issquarefree(p+i)==i%2, return(0))); 1
a(n)=forprime(p=2,, if(has(p,n), return(p))) \\ Charles R Greathouse IV, Apr 27 2015

Extensions

a(3) corrected, a(6)-a(7) added by Charles R Greathouse IV, Apr 27 2015