cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 13 results. Next

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

Original entry on oeis.org

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

Views

Author

Max Alekseyev, Jan 21 2010

Keywords

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

Programs

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

Formula

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

Extensions

Edited by Max Alekseyev, Mar 09 2010

A194366 Nonsquare positive integers n such that the fundamental unit of quadratic field Q(sqrt(n)) has norm 1 and can be written as x+y*sqrt(d) with integers x, y where d is the squarefree part of n.

Original entry on oeis.org

3, 6, 7, 11, 12, 14, 15, 19, 22, 23, 24, 27, 28, 30, 31, 33, 34, 35, 38, 39, 42, 43, 44, 46, 47, 48, 51, 54, 55, 56, 57, 59, 60, 62, 63, 66, 67, 70, 71, 75, 76, 78, 79, 83, 86, 87, 88, 91, 92, 94, 95, 96, 99, 102, 103, 105, 107, 108, 110, 111, 112, 114, 115
Offset: 1

Views

Author

Artur Jasinski, Oct 10 2011

Keywords

Comments

This sequence is a subsequence of A087643.

Examples

			35 belongs to this sequence because x^2 + 35*y^2 = 1 has the integer solution x=6, y=1.

Crossrefs

Cf. A087643, A172000.

Programs

  • 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, k1 = Max[Denominator[d1], Denominator[d2]];  If[k1 == 1, AppendTo[cr, n]]]], {n, 2, 100}]; cr

Extensions

Definition clarified by Emmanuel Vantieghem, Mar 06 2017

A197115 Nonsquare positive integers k such that the fundamental unit of the quadratic field Q(sqrt(k)) has norm -1 and can be written as x + y*sqrt(d) with integers x, y where d is the squarefree part of k.

Original entry on oeis.org

2, 8, 10, 17, 18, 26, 32, 37, 40, 41, 50, 58, 65, 68, 72, 73, 74, 82, 89, 90, 97, 98, 101, 104, 106, 113, 122, 128, 130, 137, 145, 148, 153, 160, 162, 164, 170, 185, 193, 197, 200, 202, 218, 226, 232, 233, 234, 241, 242, 250, 257, 260, 265, 269, 272, 274
Offset: 1

Views

Author

Artur Jasinski, Oct 10 2011

Keywords

Comments

This sequence is a subsequence of A172000.

Crossrefs

Cf. A087643, A172000, A194366.

Programs

  • 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, k1 = Max[Denominator[d1], Denominator[d2]];   If[k1 == 1, AppendTo[cr, n]]]], {n, 2, 400}]; cr

Extensions

Definition clarified by Emmanuel Vantieghem, Mar 06 2017

A197127 Nonsquare positive integers n such that the fundamental unit of quadratic field Q(sqrt(d))is singular.

Original entry on oeis.org

6, 14, 22, 30, 34, 38, 42, 46, 54, 56, 62, 66, 69, 70, 78, 86, 87, 93, 94, 102, 110, 114, 115, 118, 126, 130, 132, 134, 138, 142, 146, 150, 154, 155, 156, 158, 159, 166, 174, 177, 178, 182, 183, 184, 185, 186, 190, 194, 198, 206, 210, 214, 220, 222, 228, 230
Offset: 1

Views

Author

Artur Jasinski, Oct 10 2011

Keywords

Comments

x^2+n*y^2=(+/-)2^s where s is 0 or 1.
Definition: Unity is singular when GCD[n,y]<>1.

Examples

			a(1)=6 because unity of quadratic field  Q(6) is 5+2*Sqrt[6] and GCD[2,6]=2 <>1.

Crossrefs

Cf. A087643, A172000, A194366, A197115, A197128.

Programs

  • Mathematica
    cr = {}; Do[If[IntegerQ[Sqrt[n]], , kk = NumberFieldFundamentalUnits[Sqrt[n]]; d1 = kk[[1]][[2]][[1]]; d2 = kk[[1]][[1]] kk[[1]][[2]][[2]]; d4 = Numerator[d2/Sqrt[n]]; If[GCD[d4, n] == 1, , AppendTo[cr, n]]], {n, 2, 330}]; cr (*Artur Jasinski*)

A197128 Nonsquare positive integers n such that the fundamental unit of quadratic field Q(sqrt(n))is not singular.

Original entry on oeis.org

2, 3, 5, 7, 8, 10, 11, 12, 13, 15, 17, 18, 19, 20, 21, 23, 24, 26, 27, 28, 29, 31, 32, 33, 35, 37, 39, 40, 41, 43, 44, 45, 47, 48, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 63, 65, 67, 68, 71, 72, 73, 74, 75, 76, 77, 79, 80, 82, 83, 84, 85, 88, 89, 90, 91, 92
Offset: 1

Views

Author

Artur Jasinski, Oct 10 2011

Keywords

Comments

x^2+n*y^2=(+/-)2^s where s is 0 or 1.
Definition: Unity is singular when GCD[n,y]<>1.

Crossrefs

Cf. A087643, A172000, A194366, A197115, A197127.

Programs

  • Mathematica
    cr = {}; Do[If[IntegerQ[Sqrt[n]], , kk = NumberFieldFundamentalUnits[Sqrt[n]]; d1 = kk[[1]][[2]][[1]]; d2 = kk[[1]][[1]] kk[[1]][[2]][[2]]; d4 = Numerator[d2/Sqrt[n]]; If[GCD[d4, n] == 1, AppendTo[cr, n]]], {n, 2, 330}]; cr

A197170 Smallest k such that the fundamental unit (x+y*w) or (x+y*w)/2 of the real quadratic field Q(sqrt(k)) obeys gcd(k,y)=n.

Original entry on oeis.org

6, 69, 248, 115, 78, 511, 1016, 603, 70, 385, 3432, 793, 238, 2655, 14224, 1241, 3186, 703, 3980, 9177, 154, 736, 456, 1825, 3172, 13959, 2884, 319, 1110, 4619, 7136, 10659, 7174, 10255, 44856, 7067, 2926, 16185, 54280, 779, 7602, 10879, 22088, 10215, 46
Offset: 2

Views

Author

Artur Jasinski, Oct 11 2011

Keywords

Comments

Conjecture: For every n such a quadratic field with minimum k exists.

Examples

			For n=2 the unit is 2*w-5 with k=6.
For n=3 the unit is (3*w+25)/2 with k=69.
For n=4 the unit is (4*w-63) with k=248.
For n=5 the unit is 105*w-1126 with k=115.
For n=7 the unit is 185290497*w-4188548960 with k=511 (and this x and y appear in A041976 and A041977).

Crossrefs

Cf. A087643, A172000, A194366, A197115, A197127, A197128.

Programs

  • Mathematica
    cr = {}; ck = {}; Do[If[IntegerQ[Sqrt[n]], , kk = NumberFieldFundamentalUnits[Sqrt[n]]; d1 = kk[[1]][[2]][[1]]; d2 = kk[[1]][[1]] kk[[1]][[2]][[2]]; d4 = Numerator[d2/Sqrt[n]]; If[GCD[d4, n] == 1, , AppendTo[ck, GCD[d4, n]];  AppendTo[cr, n]]], {n, 2, 200000}]; aa = {}; Do[AppendTo[aa, cr[[First[Position[ck, n]][[1]]]]], {n, 2, 99}]; aa

Formula

k = A197127(m) where m is the smallest m such that A197169(m)=n.

A172001 Nonsquare positive integers n such that Pell equation y^2 - n*x^2 = -1 has rational solutions but the norm of fundamental unit of quadratic field Q(sqrt(n)) is 1.

Original entry on oeis.org

34, 136, 146, 178, 194, 205, 221, 305, 306, 377, 386, 410, 466, 482, 505, 514, 544, 545, 562, 584, 674, 689, 706, 712, 745, 776, 793, 802, 820, 850, 866, 884, 890, 898, 905, 1154, 1186, 1202, 1205, 1220, 1224, 1234, 1282, 1314, 1345, 1346, 1394, 1405, 1469
Offset: 1

Views

Author

Max Alekseyev, Jan 21 2010

Keywords

Comments

If the fundamental unit y0 + x0*sqrt(n) of Q(sqrt(n)) has norm -1, then (x0,y0) represents a rational solution to Pell equation y^2 - n*x^2 = -1. For n in this sequence, rational solutions exist but not delivered by the fundamental unit.

Crossrefs

Set difference of A000415 and its subsequence A172000.
Set difference of A087643 and its subsequence A022544.
Squarefree terms form A031398.
Odd terms form A249052.
Cf. A031399, A137351.

Formula

A positive integer n is in this sequence iff its squarefree core A007913(n) belongs to A031398.

Extensions

Edited by Max Alekseyev, Mar 09 2010

A087642 Sequence of squarefree n such that Q(sqrt(n)) has no element with a fully periodical continued fraction of period 1.

Original entry on oeis.org

3, 6, 7, 11, 14, 15, 19, 21, 22, 23, 30, 31, 33, 34, 35, 38, 39, 42, 43, 46, 47, 51, 55, 57, 59, 62, 66, 67, 69, 70, 71, 77, 78, 79, 83, 86, 87, 91, 93, 94, 95, 102, 103, 105, 107, 110, 111, 114, 115, 118, 119, 123, 127, 129, 131, 133, 134, 138, 139, 141, 142, 143, 146
Offset: 3

Views

Author

Thomas Baruchel, Sep 16 2003

Keywords

Comments

Diophantine equation x^2 - n.y^2 + 4 = 0 has no solution (x,y) for a given squarefree n. Squarefree n not in the sequence A013946. Same sequence with square factors allowed is A087643.

Examples

			3 is in the sequence because no [k,k,k,k,...] is in Q(sqrt(3))
5 is not in the sequence since Q(sqrt(5)) contains [1,1,1,1,...]

Crossrefs

Cf. A087643, A013946.

A197169 Values of gcd(n,y) for successive y = A197128(n).

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 46, 2, 2, 2, 2, 3, 10, 6, 2, 3, 3, 2, 2, 2, 6, 5, 2, 2, 5, 2, 2, 2, 2, 2, 2, 22, 5, 2, 2, 3, 2, 2, 3, 2, 2, 3, 46, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 14, 2, 4, 3, 2, 2, 2, 4, 14, 3, 2, 3, 2, 5, 2, 2, 2, 5, 2, 2, 2, 3, 6, 29, 3, 2, 2, 3
Offset: 1

Views

Author

Artur Jasinski, Oct 11 2011

Keywords

Crossrefs

Cf. A087643, A172000, A194366, A197115, A197127, A197128.

Programs

  • Mathematica
    cr = {}; Do[If[IntegerQ[Sqrt[n]], , kk = NumberFieldFundamentalUnits[Sqrt[n]]; d1 = kk[[1]][[2]][[1]]; d2 = kk[[1]][[1]] kk[[1]][[2]][[2]]; d4 = Numerator[d2/Sqrt[n]]; If[GCD[d4, n] == 1, , AppendTo[cr, GCD[d4, n]]]], {n, 2, 20000}]; cr

A197171 Values k such that singular quadratic unity of Q(k) have gcd(k,y) = 2.

Original entry on oeis.org

6, 14, 22, 30, 34, 38, 42, 54, 56, 62, 66, 86, 94, 102, 110, 118, 126, 132, 134, 138, 142, 146, 150, 156, 158, 166, 174, 178, 182, 186, 190, 194, 198, 206, 210, 214, 220, 222, 228, 230, 246, 254, 258, 262, 270, 278, 282, 286, 294, 302, 306, 310, 322, 326
Offset: 2

Views

Author

Artur Jasinski, Oct 11 2011

Keywords

Comments

Conjecture: This sequence is infinite.

Crossrefs

Cf. A087643, A172000, A194366, A197115, A197127, A197128, A197170.

Programs

  • Mathematica
    cr = {}; ck = {}; Do[If[IntegerQ[Sqrt[n]], , kk = NumberFieldFundamentalUnits[Sqrt[n]]; d1 = kk[[1]][[2]][[1]]; d2 = kk[[1]][[1]] kk[[1]][[2]][[2]]; d4 = Numerator[d2/Sqrt[n]]; If[GCD[d4, n] == 1, , AppendTo[ck, GCD[d4, n]]; AppendTo[cr, n]]], {n, 2, 200000}];aa = {}; Do[If[ck[[n]] == 2, AppendTo[aa, cr[[n]]]], {n, 1, Length[cr]}]; aa
Showing 1-10 of 13 results. Next

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