

A172000


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


18



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
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OFFSET

1,1


COMMENTS

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


LINKS

Table of n, a(n) for n=1..60.


FORMULA

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


MATHEMATICA

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 *)


PROG

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


CROSSREFS

Sequence in context: A025284 A140328 A000415 * A096691 A202057 A263651
Adjacent sequences: A171997 A171998 A171999 * A172001 A172002 A172003


KEYWORD

nonn


AUTHOR

Max Alekseyev, Jan 21 2010


EXTENSIONS

Edited by Max Alekseyev, Mar 09 2010


STATUS

approved



