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A166687
Numbers of the form x^2 + y^2 + 1, x, y integers.
4
1, 2, 3, 5, 6, 9, 10, 11, 14, 17, 18, 19, 21, 26, 27, 30, 33, 35, 37, 38, 41, 42, 46, 50, 51, 53, 54, 59, 62, 65, 66, 69, 73, 74, 75, 81, 82, 83, 86, 90, 91, 98, 99, 101, 102, 105, 107, 110, 114, 117, 118, 122, 123, 126, 129, 131, 137, 138, 145, 146, 147, 149, 150, 154, 158, 161
OFFSET
1,2
COMMENTS
A001481 is the main entry for this sequence.
As Ng points out (Lemma 2.2), each prime divides some member of this sequence: 2 divides a(2) = 2, 3 divides a(3) = 3, 5 divides a(4) = 5, 7 divides a(9) = 14, etc. - Charles R Greathouse IV, Jan 04 2016
LINKS
Jia Hong Ray Ng, Quarternions and the four square theorem, 2008 Summer VIGRE Program for Undergraduates
Yu-Chen Sun and Hao Pan, The Green-Tao theorem for primes of the form x^2 + y^2 + 1, Monatshefte für Mathematik vol. 189 (2019), pp. 715-733. arXiv:1708.08629 [math.NT]
MAPLE
N:= 1000: # to get all terms <= N
S:= {seq(seq(x^2+y^2+1, y=0..floor(sqrt(N-1-x^2))), x=0..floor(sqrt(N-1)))}:
sort(convert(S, list)); # Robert Israel, Jan 05 2016
MATHEMATICA
Select[Range@ 162, Resolve[Exists[{x, y}, Reduce[# == x^2 + y^2 + 1, {x, y}, Integers]]] &] (* Michael De Vlieger, Jan 05 2016 *)
PROG
(PARI) is(n)=my(f=factor(n-1)); for(i=1, #f~, if(f[i, 1]%4==3 && f[i, 2]%2, return(0))); 1 \\ Charles R Greathouse IV, Jan 04 2016
(PARI) list(lim)=my(v=List(), t); lim\=1; for(m=0, sqrtint(lim-1), t=m^2+1; for(n=0, min(sqrtint(lim-t), m), listput(v, t+n^2))); Set(v) \\ Charles R Greathouse IV, Jan 05 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Mar 05 2010
STATUS
approved