A293564 Starts of a record number of consecutive integers n such that n^2 + 1 is composite.

3, 7, 27, 41, 95, 185, 351, 497, 3391, 3537, 45371, 82735, 99065, 357165, 840905, 3880557, 27914937, 40517521, 104715207, 1126506905, 2084910531, 2442825347, 4332318177, 6716598047, 17736392221, 18205380337, 30869303807, 68506021365, 78491213265, 85620067845

Offset: 1

Author

Amiram Eldar, Oct 12 2017

Keywords

• nonn

Comments

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.

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, ...

Examples

			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.
		

Links

• Betty Garrison, Consecutive integers for which n^2+1 is composite, Pacific Journal of Mathematics, Vol. 97, No. 1 (1981), pp. 93-96.

Crossrefs

Cf. A002496, A002522, A005574.

Programs

Mathematica

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
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 *)

Extensions

a(17)-a(20) from Robert G. Wilson v, Oct 12 2017

a(21)-a(22) from Giovanni Resta, Oct 13 2017

a(23)-a(27) from Chai Wah Wu, May 16 2018

a(28)-a(30) from Giovanni Resta, May 18 2018