

A257108


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


5



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

0,1


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)


LINKS

Table of n, a(n) for n=0..11.


FORMULA

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


EXAMPLE

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


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 *)


PROG

(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


CROSSREFS

Cf. A020754.
Sequence in context: A030090 A260298 A283807 * A274385 A075461 A059785
Adjacent sequences: A257105 A257106 A257107 * A257109 A257110 A257111


KEYWORD

nonn,more


AUTHOR

JuriStepan Gerasimov, Apr 23 2015


EXTENSIONS

a(8) from Robert Israel, Apr 23 2015
a(9)a(11) from Charles R Greathouse IV, Apr 29 2015


STATUS

approved



