A257108 Smallest prime p such that none of p+1, p+2, ..., p+n are squarefree.

2, 3, 7, 47, 241, 2887, 57119, 217069, 37923937, 211014919, 221167421, 221167421

Offset: 0

Author

Juri-Stepan Gerasimov, Apr 23 2015

Keywords

• nonn
• more

Comments

a(10) = 221167421.

From Robert Israel, Apr 23 2015: (Start)

a(n) >= A020754(n), with equality when A020754(n) is prime. This occurs for n = 2,3,4,5,8 and 11.

Each a(n) exists: given distinct primes q_j, j=1..n, such that q_j does not divide j, by Dirichlet's theorem there is some prime in the arithmetic progression

{x: x == -j (mod q_j^2) for j=1..n}.

(End)

Examples

			47 is a(3) because none of 2^2*12 = 48, 7^2 = 49, 2*5^2 = 50 is squarefree.
		

Crossrefs

Cf. A020754.

Programs

Maple

p:= 2:
A[0]:= 2:
for n from 1 to 8 do
  while ormap(numtheory:-issqrfree, [seq(p+i,i=1..n)]) do p:= nextprime(p) od:
  A[n]:= p;
od:
seq(A[i],i=1..8); # Robert Israel, Apr 23 2015

Mathematica

lst={2};Do[If[Union[SquareFreeQ/@Range[Prime[n]+1,Prime[n]+Length[lst]]]=={False},AppendTo[lst,Prime[n]]],{n,10^5}];lst (* Ivan N. Ianakiev, May 02 2015 *)

PARI

a(n)=forprime(p=2, , for(k=1, n, if(issquarefree(p+k), next(2))); return(p)) \\ Charles R Greathouse IV, Apr 29 2015

Formula

a(n) << A002110(n)^10 by the CRT and Xylouris' improvement to Linnik's theorem. - Charles R Greathouse IV, Apr 29 2015

Extensions

a(8) from Robert Israel, Apr 23 2015

a(9)-a(11) from Charles R Greathouse IV, Apr 29 2015