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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 (list; graph; refs; listen; history; text; internal format)
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

Juri-Stepan 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

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Last modified December 9 22:27 EST 2019. Contains 329880 sequences. (Running on oeis4.)