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 12 results. Next

A121085 Number of primitive Pythagorean-like triples a^2+b^2=c^2+k for k=-3 with 0

Original entry on oeis.org

3, 30, 293, 2881, 28871, 288685, 2886366, 28868362, 288673693, 2886752763
Offset: 1

Views

Author

Tito Piezas III, Aug 11 2006

Keywords

Comments

It is conjectured by the first author that a(n)/10^n as n->inf is 1/(2*sqrt(3)) = 0.28867...

Examples

			a(1)=3 because there are 3 solutions (a,b,c) as (2,3,4),(5,6,8),(4,9,10) with 0<c<=10^1.

Crossrefs

Cf. A101931, A121082, A121083, A121084, A121086, A121087, A121088

Programs

  • Mathematica
    countTriples[m_, k_] := Module[ {c2, c2odd, total = 0, fax, g}, Do[ c2 = c^2 + k; If[c2 < 2, Continue[]]; c2odd = c2; While[EvenQ[c2odd], c2odd /= 2]; If [c2odd==1, If [OddQ[Log[2,c2]], total++ ]; Continue[]]; If[Mod[c2odd, 4] == 3, Continue[]]; g = GCD[c2odd, 100947]; If[g != 1 && g^2 != GCD[c2odd, 10190296809], Continue[]]; fax = Map[{Mod[ #[[1]],4],#[[2]]}&, FactorInteger[c2odd]]; If[Apply[Or, Map[ #[[1]] == 3 && OddQ[ #[[2]]] &, fax]], Continue []]; fax = Cases[fax, {1,aa_}:>aa+1]; fax = Ceiling[Apply[Times,fax]/2]; total += fax;, {c,m}]; total] (* Courtesy of Daniel Lichtblau of Wolfram Research *)

Extensions

First few terms found by Tito Piezas III, James Waldby (j-waldby(AT)pat7.com). Subsequent terms found by Andrzej Kozlowski (akoz(AT)mimuw.edu.pl), Daniel Lichtblau (danl(AT)wolfram.com).
a(7) from Max Alekseyev, Jul 03 2011
a(8)-a(10) from Hiroaki Yamanouchi, Oct 17 2015

A121082 Number of primitive Pythagorean-like triples a^2+b^2=c^2+k for k=-1 with c<=10^n.

Original entry on oeis.org

2, 14, 126, 1238, 12517, 124973, 1249931, 12500186, 125000681, 1250005179
Offset: 1

Views

Author

Tito Piezas III, Aug 11 2006

Keywords

Comments

It is conjectured by the first author that a(n)/10^n as n->inf is 1/8 = 0.125.

Examples

			a(1)=2 because there are 2 solutions (a,b,c) as (2,2,3),(4,8,9) with c<=10^1.

Crossrefs

Cf. A101931.

Programs

  • Mathematica
    Courtesy of Daniel Lichtblau of Wolfram Research: countTriples[m_, k_] := Module[ {c2, c2odd, total = 0, fax, g}, Do[ c2 = c^2 + k; If[c2 < 2, Continue[]]; c2odd = c2; While[EvenQ[c2odd], c2odd /= 2]; If [c2odd==1, If [OddQ[Log[2,c2]], total++ ]; Continue[]]; If[Mod[c2odd, 4] == 3, Continue[]]; g = GCD[c2odd, 100947]; If[g != 1 && g^2 != GCD[c2odd, 10190296809], Continue[]]; fax = Map[{Mod[ #[[1]],4],#[[2]]}&, FactorInteger[c2odd]]; If[Apply[Or, Map[ #[[1]] == 3 && OddQ[ #[[2]]] &, fax]], Continue []]; fax = Cases[fax, {1,aa_}:>aa+1]; fax = Ceiling[Apply[Times,fax]/2]; total += fax;, {c,m}]; total]

Extensions

First few terms found by Tito Piezas III, James Waldby (j-waldby(AT)pat7.com). Subsequent terms found by Andrzej Kozlowski (akoz(AT)mimuw.edu.pl), Daniel Lichtblau (danl(AT)wolfram.com).
a(8)-a(10) from Hiroaki Yamanouchi, Oct 17 2015

A121083 Number of primitive Pythagorean-like triples a^2+b^2=c^2+k for k=-2 with 0

Original entry on oeis.org

3, 19, 182, 1779, 17697, 176794, 1768021, 17676780, 176776851, 1767763756
Offset: 1

Views

Author

Tito Piezas III, Aug 11 2006

Keywords

Comments

It is conjectured by the first author that a(n)/10^n as n->inf is 1/(4*sqrt(2)) = 0.17677...

Examples

			a(1)=3 because there are 3 solutions (a,b,c) as (1,1,2), (3,5,6), (7,7,10) with 0<c<=10^1.

Crossrefs

Cf. A101931.

Programs

  • Mathematica
    countTriples[m_, k_] := Module[ {c2, c2odd, total = 0, fax, g}, Do[ c2 = c^2 + k; If[c2 < 2, Continue[]]; c2odd = c2; While[EvenQ[c2odd], c2odd /= 2]; If [c2odd==1, If [OddQ[Log[2,c2]], total++ ]; Continue[]]; If[Mod[c2odd, 4] == 3, Continue[]]; g = GCD[c2odd, 100947]; If[g != 1 && g^2 != GCD[c2odd, 10190296809], Continue[]]; fax = Map[{Mod[ #[[1]],4],#[[2]]}&, FactorInteger[c2odd]]; If[Apply[Or, Map[ #[[1]] == 3 && OddQ[ #[[2]]] &, fax]], Continue []]; fax = Cases[fax, {1,aa_}:>aa+1]; fax = Ceiling[Apply[Times,fax]/2]; total += fax;, {c,m}]; total] (* Courtesy of Daniel Lichtblau of Wolfram Research *)

Extensions

First few terms found by Tito Piezas III, James Waldby (j-waldby(AT)pat7.com). Subsequent terms found by Andrzej Kozlowski (akoz(AT)mimuw.edu.pl), Daniel Lichtblau (danl(AT)wolfram.com).
a(8)-a(10) from Hiroaki Yamanouchi, Oct 17 2015

A121084 Number of primitive Pythagorean-like triples a^2+b^2=c^2+k for k=2 with 0

Original entry on oeis.org

1, 10, 100, 983, 9912, 99211, 991714, 9918739, 99187754, 991897081
Offset: 1

Views

Author

Tito Piezas III, Aug 11 2006

Keywords

Examples

			a(1)=1 because there is one solution (a,b,c) as (3,3,4) with 0<c<=10^1.

Crossrefs

Cf. A101931.

Programs

  • Mathematica
    Courtesy of Daniel Lichtblau of Wolfram Research: countTriples[m_, k_] := Module[ {c2, c2odd, total = 0, fax, g}, Do[ c2 = c^2 + k; If[c2 < 2, Continue[]]; c2odd = c2; While[EvenQ[c2odd], c2odd /= 2]; If [c2odd==1, If [OddQ[Log[2,c2]], total++ ]; Continue[]]; If[Mod[c2odd, 4] == 3, Continue[]]; g = GCD[c2odd, 100947]; If[g != 1 && g^2 != GCD[c2odd, 10190296809], Continue[]]; fax = Map[{Mod[ #[[1]],4],#[[2]]}&, FactorInteger[c2odd]]; If[Apply[Or, Map[ #[[1]] == 3 && OddQ[ #[[2]]] &, fax]], Continue []]; fax = Cases[fax, {1,aa_}:>aa+1]; fax = Ceiling[Apply[Times,fax]/2]; total += fax;, {c,m}]; total]

Extensions

First few terms found by Tito Piezas III, James Waldby (j-waldby(AT)pat7.com). Subsequent terms found by Andrzej Kozlowski (akoz(AT)mimuw.edu.pl), Daniel Lichtblau (danl(AT)wolfram.com).
a(7) from Max Alekseyev, May 30 2007
a(8)-a(10) from Asif Ahmed, Dec 07 2024

A121087 Number of primitive Pythagorean-like triples a^2+b^2=c^2+k for k=-5 with 0

Original entry on oeis.org

1, 22, 223, 2217, 22354, 223667, 2235713, 22360389, 223610157
Offset: 1

Views

Author

Tito Piezas III, Aug 11 2006

Keywords

Comments

It is conjectured by the first author that a(n)/10^n as n->inf is 1/(2*sqrt(5)) = 0.22360...

Examples

			a(1)=1 because there is one solution (a,b,c) as (2,4,5) with 0<c<=10^1.

Crossrefs

Cf. A101931.

Programs

  • Mathematica
    (* Courtesy of Daniel Lichtblau of Wolfram Research *)
countTriples[m_, k_] := Module[ {c2, c2odd, total = 0, fax, g}, Do[ c2 = c^2 + k; If[c2 < 2, Continue[]]; c2odd = c2; While[EvenQ[c2odd], c2odd /= 2]; If [c2odd==1, If [OddQ[Log[2,c2]], total++ ]; Continue[]]; If[Mod[c2odd, 4] == 3, Continue[]]; g = GCD[c2odd, 100947]; If[g != 1 && g^2 != GCD[c2odd, 10190296809], Continue[]]; fax = Map[{Mod[ #[[1]],4],#[[2]]}&, FactorInteger[c2odd]]; If[Apply[Or, Map[ #[[1]] == 3 && OddQ[ #[[2]]] &, fax]], Continue []]; fax = Cases[fax, {1,aa_}:>aa+1]; fax = Ceiling[Apply[Times,fax]/2]; total += fax;, {c,m}]; total]

Extensions

First few terms found by Tito Piezas III, James Waldby (j-waldby(AT)pat7.com)
Subsequent terms found by Andrzej Kozlowski (akoz(AT)mimuw.edu.pl), Daniel Lichtblau (danl(AT)wolfram.com)
a(7) from Max Alekseyev, May 30 2007
a(8)-a(9) from Lars Blomberg, Dec 22 2015

A121086 Number of primitive Pythagorean-like triples a^2+b^2=c^2+k for k=3 with 0

Original entry on oeis.org

1, 13, 119, 1219, 12115, 121054, 1210480, 12101765, 121011208, 1210128842
Offset: 1

Views

Author

Tito Piezas III, Aug 11 2006

Keywords

Examples

			a(1)=1 because there is one solution (a,b,c) as (4,6,7) with 0<c<=10^1.

Crossrefs

Cf. A101931, A121082, A121083, A121084, A121085, A121087, A121088

Programs

  • Mathematica
    (* Courtesy of Daniel Lichtblau of Wolfram Research *)
countTriples[m_, k_] := Module[ {c2, c2odd, total = 0, fax, g}, Do[ c2 = c^2 + k; If[c2 < 2, Continue[]]; c2odd = c2; While[EvenQ[c2odd], c2odd /= 2]; If [c2odd==1, If [OddQ[Log[2,c2]], total++ ]; Continue[]]; If[Mod[c2odd, 4] == 3, Continue[]]; g = GCD[c2odd, 100947]; If[g != 1 && g^2 != GCD[c2odd, 10190296809], Continue[]]; fax = Map[{Mod[ #[[1]],4],#[[2]]}&, FactorInteger[c2odd]]; If[Apply[Or, Map[ #[[1]] == 3 && OddQ[ #[[2]]] &, fax]], Continue []]; fax = Cases[fax, {1,aa_}:>aa+1]; fax = Ceiling[Apply[Times,fax]/2]; total += fax;, {c,m}]; total]

Extensions

First few terms found by Tito Piezas III, James Waldby (j-waldby(AT)pat7.com)
Subsequent terms found by Andrzej Kozlowski (akoz(AT)mimuw.edu.pl), Daniel Lichtblau (danl(AT)wolfram.com)
a(7) from Max Alekseyev, Jul 04 2011
a(8)-a(9) from Lars Blomberg, Dec 22 2015
a(10) from Asif Ahmed, Dec 07 2024

A121088 Number of primitive Pythagorean-like triples a^2+b^2=c^2+k for k=5 with 0

Original entry on oeis.org

1, 20, 202, 2046, 20589, 205489, 2055224, 20551650, 205500435, 2055052214
Offset: 1

Views

Author

Tito Piezas III, Aug 11 2006

Keywords

Examples

			a(1)=1 because there is one solution (a,b,c) as (4,5,6) with 0<c<=10^1.

Crossrefs

Cf. A101931, A121082, A121083, A121084, A121085, A121086, A121087

Programs

  • Mathematica
    (* Courtesy of Daniel Lichtblau of Wolfram Research *)
countTriples[m_, k_] := Module[ {c2, c2odd, total = 0, fax, g}, Do[ c2 = c^2 + k; If[c2 < 2, Continue[]]; c2odd = c2; While[EvenQ[c2odd], c2odd /= 2]; If [c2odd==1, If [OddQ[Log[2,c2]], total++ ]; Continue[]]; If[Mod[c2odd, 4] == 3, Continue[]]; g = GCD[c2odd, 100947]; If[g != 1 && g^2 != GCD[c2odd, 10190296809], Continue[]]; fax = Map[{Mod[ #[[1]],4],#[[2]]}&, FactorInteger[c2odd]]; If[Apply[Or, Map[ #[[1]] == 3 && OddQ[ #[[2]]] &, fax]], Continue []]; fax = Cases[fax, {1,aa_}:>aa+1]; fax = Ceiling[Apply[Times,fax]/2]; total += fax;, {c,m}]; total]

Extensions

First few terms found by Tito Piezas III, James Waldby (j-waldby(AT)pat7.com)
Subsequent terms found by Andrzej Kozlowski (akoz(AT)mimuw.edu.pl), Daniel Lichtblau (danl(AT)wolfram.com)
a(6) corrected and a(7) added by Max Alekseyev, Jul 04 2011
a(8)-a(9) from Lars Blomberg, Dec 22 2015
a(10) from Asif Ahmed, Dec 07 2024

A239581 Number of primitive Pythagorean triangles (x, y, z) with legs x < y < 10^n.

Original entry on oeis.org

1, 18, 179, 1788, 17861, 178600, 1786011, 17860355, 178603639, 1786036410, 17860362941
Offset: 1

Views

Author

Martin Renner, Mar 26 2014

Keywords

Comments

A Pythagorean triangle is a right triangle with integer side lengths x, y, z forming a Pythagorean triple (x, y, z). It is called primitive, if gcd(x, y, z) = 1.
Because (x, y, z) is equivalent to (y, x, z), the total number of primitive Pythagorean triangles with legs x, y < 10^n is b(n) = 2*a(n) = 2, 36, 358, 3576, 35722, ...

Examples

			a(1) = 1, because the only primitive Pythagorean triangle with x < y < 10 is [3, 4, 5].

Links

Crossrefs

Cf. A008846, A020882, A020883, A020884, A046086, A046087, A101931, A101929, A239744, A239786.

Extensions

a(6)-a(11) from Giovanni Resta, Mar 27 2014

A239744 Number of Pythagorean triangles (x, y, z) with legs x < y <= 10^n.

Original entry on oeis.org

2, 63, 1034, 14474, 185864, 2269788, 26809924, 309224756, 3503496007, 39147452729, 432599522197
Offset: 1

Views

Author

Martin Renner, Mar 26 2014

Keywords

Comments

A Pythagorean triangle is a right triangle with integer side lengths x, y, z forming a Pythagorean triple (x, y, z).
Because (x, y, z) is equivalent to (y, x, z), the total number of Pythagorean triangles with legs x, y < 10^n is b(n) = 2*a(n) = 4, 126, 2068, 28948, 371728, ...

Examples

			a(1) = 2, because the only two Pythagorean triangles with x < y < 10 are [3, 4, 5] and [6, 8, 10].

Links

Crossrefs

Cf. A008846, A020882, A020883, A020884, A046086, A046087, A101931, A101929, A239581, A239786.

Extensions

a(6)-a(11) from Giovanni Resta, Mar 27 2014

A239786 Number of Pythagorean triangles (x, y, z) with legs x < y < 10^n.

Original entry on oeis.org

2, 62, 1032, 14471, 185860, 2269783, 26809918, 309224749, 3503495999, 39147452720, 432599522187
Offset: 1

Views

Author

Martin Renner, Mar 26 2014

Keywords

Comments

A Pythagorean triangle is a right triangle with integer side length x, y, z forming a Pythagorean triple (x, y, z).
Because (x, y, z) is equivalent to (y, x, z), the total number of Pythagorean triangles with legs x, y < 10^n is b(n) = 2*a(n) = 4, 124, 2064, 28942, ...

Links

Crossrefs

Cf. A008846, A020882, A020883, A020884, A046086, A046087, A101931, A101929, A239581, A239744.

Extensions

a(5)-a(11) from Giovanni Resta, Mar 27 2014
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