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

1,1

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, ...

Links

Table of n, a(n) for n=1..30.

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

Example

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.

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 *)

Crossrefs

Cf. A002496, A002522, A005574.

Sequence in context: A175198 A272530 A225038 * A056257 A066021 A259595

Adjacent sequences: A293561 A293562 A293563 * A293565 A293566 A293567

Keyword

nonn

Author

Amiram Eldar, 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

Status

approved