A172001 -id:A172001 - cp's OEIS frontend

A172000 Nonsquare positive integers n such that the fundamental unit of quadratic field Q(sqrt(n)) has norm -1.

2, 5, 8, 10, 13, 17, 18, 20, 26, 29, 32, 37, 40, 41, 45, 50, 52, 53, 58, 61, 65, 68, 72, 73, 74, 80, 82, 85, 89, 90, 97, 98, 101, 104, 106, 109, 113, 116, 117, 122, 125, 128, 130, 137, 145, 148, 149, 153, 157, 160, 162, 164, 170, 173, 180, 181, 185, 193, 197, 200

Offset: 1

Max Alekseyev, Jan 21 2010

nonn

Complement of A087643 in the nonsquare integers A000037.
Subsequence of A000415, their set difference form A172001.
Contains A003814 as a subsequence, their squarefree terms coincide and form A003654.
It seems that this sequence also gives the values of n such that the equation x^2 - n*y^2 = n has integer solutions. - Colin Barker, Aug 20 2013

cr = {}; Do[If[IntegerQ[Sqrt[n]], , kk = NumberFieldFundamentalUnits[Sqrt[n]]; d1 = kk[[1]][[2]][[1]]; d2 = kk[[1]][[1]] kk[[1]][[2]][[2]]; d3 = Expand[(d1 + d2) (d1 - d2)]; If[d3 == -1, AppendTo[cr, n]]], {n, 2, 1000}]; cr (* Artur Jasinski, Oct 10 2011 *)

{ for(n=1,1000, if(issquare(n),next); if( norm(bnfinit(x^2-n).fu[1])==-1, print1(n,", ")) ) }

A positive integer n is in this sequence iff its squarefree core A007913(n) belongs to A003654.

Edited by Max Alekseyev, Mar 09 2010

A249052 Odd numbers n without prime factors of the form 4k+3 such that x^2-n*y^2=-1 has no solutions.

Original entry on oeis.org

205, 221, 305, 377, 505, 545, 689, 725, 745, 793, 905, 1205, 1345, 1405, 1469, 1513, 1517, 1537, 1717, 1885, 1945, 1961, 2005, 2041, 2045, 2105, 2225, 2245, 2329, 2353, 2525, 2533, 2669, 2701, 2845, 2993, 3005, 3205, 3305, 3497, 3505, 3737, 3757, 3805, 3893, 3965, 4069, 4105, 4145, 4205, 4321, 4369

Offset: 1

Joerg Arndt, Apr 09 2008

nonn

A positive odd integer n is in this sequence iff its squarefree core A007913(n) belongs to A031398. - Max Alekseyev, Jun 26 2022

Odd terms of A172001.

Cf. A031398, A031399, A137351.

cp's OEIS Frontend

A172000 Nonsquare positive integers n such that the fundamental unit of quadratic field Q(sqrt(n)) has norm -1.

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A249052 Odd numbers n without prime factors of the form 4k+3 such that x^2-n*y^2=-1 has no solutions.

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