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