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A373618 Least prime starting a run of n consecutive primes p_i, i=1..n, such that p_i + 1 is squarefree and p_(n+1) + 1 is not squarefree.

%I A373618 #50 Jul 07 2024 13:01:52
%S A373618 2,37,397,389,11617,11597,11593,2048509,2772409,5193997,33933701,
%T A373618 125624813,125624809,432787781,432787777,4762221193,4762221181,
%U A373618 182839149373,547414016069,551900822513
%N A373618 Least prime starting a run of n consecutive primes p_i, i=1..n, such that p_i + 1 is squarefree and p_(n+1) + 1 is not squarefree.
%e A373618 a(1) = 2, because 2 is the least prime starting a run of 1 prime such that 2+1 is squarefree and 3+1 = 4 = 2^2 is not squarefree.
%e A373618 For n=4 the first run of 4 squarefree p+1 starts at a(4) = 389, and no run of n=3 so a(3) = 397 is the ending 3 of this run.
%e A373618   p              = 389, 397, 401, 409, 419
%e A373618   p+1 squarefree = yes  yes  yes  yes  no
%e A373618   n=4 run          \----------------/
%e A373618   n=3 run               \-----------/
%t A373618 a[n_]:=Module[{k=1}, While[pr=Product[Boole[SquareFreeQ[Prime[k+i-1]+1]], {i, n}]==0||pr&& Boole[SquareFreeQ[Prime[k+n]+1]]==1, k++]; Prime[k]]; Array[a, 8] (* _Stefano Spezia_, Jun 11 2024 *)
%Y A373618 Cf. A005117, A072875, A086560, A369097.
%K A373618 nonn,more
%O A373618 1,1
%A A373618 _Jean-Marc Rebert_, Jun 11 2024