%I
%S 3,7,27,41,95,185,351,497,3391,3537,45371,82735,99065,357165,840905,
%T 3880557,27914937,40517521,104715207,1126506905,2084910531,2442825347,
%U 4332318177,6716598047,17736392221,18205380337,30869303807,68506021365,78491213265,85620067845
%N Starts of a record number of consecutive integers n such that n^2 + 1 is composite.
%C Garrison proved in 1981 that there are arbitrarily long strings of consecutive integers n such that n^2 + 1 is composite. Thus, if the sequence of primes of the form n^2 + 1 (A002496) is infinite, this sequence is also infinite.
%C The record lengths are 1, 3, 9, 13, 15, 19, 33, 39, 45, 87, 99, 111, 129, 151, 211, 287, 329, 345, 443, 501, 525, 533, 563, 579, 613, 623, 633, 635, 639, 689, ...
%H Betty Garrison, <a href="https://msp.org/pjm/1981/971/pjmv97n1p08s.pdf">Consecutive integers for which n^2+1 is composite</a>, Pacific Journal of Mathematics, Vol. 97, No. 1 (1981), pp. 9396.
%e 7 is in the sequence since 7^2+1, 8^2+1 and 9^2+1 are composites, the first string of 3 consecutive composite numbers of the form n^2 + 1.
%t aQ[n_] := PrimeQ[n^2 + 1]; s = Flatten[Position[Range[100], _?(aQ[#] &)]]; dm = 1; a = {}; For[k = 0, k < Length[s]  1, k++; d = s[[k + 1]]s[[k]]; If[d > dm, dm = d; AppendTo[a, s[[k]] + 1]]]; a
%t f[n_] := f[n] = Block[{s, k = f[n 1]}, s = Boole@ PrimeQ[ Range[k, k +n 1]^2 +1]; While[Plus @@ s > 0, s = Join[s, Boole@ PrimeQ[{(k +n)^2 + 1, (k +n +1)^2 +1}]]; s = Drop[s, 2]; k += 2]; k]; f[1] = 3; Do[ Print[{n, f@n}], {n, 329}] (* _Robert G. Wilson v_, Oct 12 2017 *)
%Y Cf. A002496, A002522, A005574.
%K nonn
%O 1,1
%A _Amiram Eldar_, Oct 12 2017
%E a(17)a(20) from _Robert G. Wilson v_, Oct 12 2017
%E a(21)a(22) from _Giovanni Resta_, Oct 13 2017
%E a(23)a(27) from _Chai Wah Wu_, May 16 2018
%E a(28)a(30) from _Giovanni Resta_, May 18 2018
