cp's OEIS Frontend

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

%I A257108 #34 Jan 22 2022 23:41:25
%S A257108 2,3,7,47,241,2887,57119,217069,37923937,211014919,221167421,221167421
%N A257108 Smallest prime p such that none of p+1, p+2, ..., p+n are squarefree.
%C A257108 a(10) = 221167421.
%C A257108 From _Robert Israel_, Apr 23 2015: (Start)
%C A257108 a(n) >= A020754(n), with equality when A020754(n) is prime.  This occurs for n = 2,3,4,5,8 and 11.
%C A257108 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
%C A257108 {x: x == -j (mod q_j^2) for j=1..n}.
%C A257108 (End)
%F A257108 a(n) << A002110(n)^10 by the CRT and Xylouris' improvement to Linnik's theorem. - _Charles R Greathouse IV_, Apr 29 2015
%e A257108 47 is a(3) because none of 2^2*12 = 48, 7^2 = 49, 2*5^2 = 50 is squarefree.
%p A257108 p:= 2:
%p A257108 A[0]:= 2:
%p A257108 for n from 1 to 8 do
%p A257108   while ormap(numtheory:-issqrfree, [seq(p+i,i=1..n)]) do p:= nextprime(p) od:
%p A257108   A[n]:= p;
%p A257108 od:
%p A257108 seq(A[i],i=1..8); # _Robert Israel_, Apr 23 2015
%t A257108 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 *)
%o A257108 (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
%Y A257108 Cf. A020754.
%K A257108 nonn,more
%O A257108 0,1
%A A257108 _Juri-Stepan Gerasimov_, Apr 23 2015
%E A257108 a(8) from _Robert Israel_, Apr 23 2015
%E A257108 a(9)-a(11) from _Charles R Greathouse IV_, Apr 29 2015