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.

User: Tito Piezas III

Tito Piezas III's wiki page.

Tito Piezas III has authored 10 sequences.

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

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

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

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

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

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

A060981 Primes of the form 4k^2 - 152k + 1487 + (6k - 114)*(-1)^k.

Original entry on oeis.org

1373, 1447, 1097, 1163, 853, 911, 641, 691, 461, 503, 313, 347, 197, 223, 113, 131, 61, 71, 41, 43, 53, 47, 97, 83, 173, 151, 281, 251, 421, 383, 593, 547, 797, 743, 1033, 971, 1301, 1231, 1601, 1523, 1933, 1847, 2297, 2203, 2693, 2591, 3121, 3011, 3581
Offset: 1

Views

Author

Tito Piezas III, May 11 2001

Keywords

Comments

Generates distinct values, the first 61 of which (i.e., those corresponding to k = 0..60) are primes.

Links

Crossrefs

Cf. A005846.

Programs

  • Mathematica
    lst={};Do[p=4*n^2-152*n+1487+(6*n-114)*(-1)^n;If[PrimeQ[p],AppendTo[lst,p]],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 28 2009 *)
Select[Table[4n^2-152n+1487+(6n-114)(-1)^n,{n,0,50}],PrimeQ] (* Harvey P. Dale, Sep 15 2011 *)
  • PARI
    { n=0; for (m=0, 10000, p=4*m^2 - 152*m + 1487 + (6*m - 114)*(-1)^m; if (isprime(p), write("b060981.txt", n++, " ", p); if (n==1000, break)); ) } \\ Harry J. Smith, Jul 15 2009

A057604 Primes of the form 4*k^2 + 163.

Original entry on oeis.org

163, 167, 179, 199, 227, 263, 307, 359, 419, 487, 563, 647, 739, 839, 947, 1063, 1187, 1319, 1459, 1607, 2099, 2467, 2663, 3079, 3299, 3527, 4007, 4259, 4519, 4787, 5347, 5639, 5939, 6247, 6563, 7219, 7559, 7907, 8263, 8627, 8999, 9767, 10163, 10567, 10979, 11399, 11827, 12263
Offset: 1

Views

Author

Tito Piezas III, Oct 08 2000

Keywords

Comments

These numbers are not prime in O_Q(sqrt(-163)). If p = n^2 + 163, then (n - sqrt(-163))*(n + sqrt(-163)) = p. - Alonso del Arte, Dec 18 2017

Links

Crossrefs

Cf. A005846, A007641, A007635, A057605.

Programs

  • Magma
    [a: n in [0..400] | IsPrime(a) where a is 4*n^2 + 163] // Vincenzo Librandi, Aug 07 2010
  • Mathematica
    Select[Table[4n^2 + 163, {n, 0, 70}], PrimeQ] (* Vincenzo Librandi, Jul 15 2012 *)
  • PARI
    lista(nn) = for(n=0, nn, my(p = 4*n^2 + 163); if(isprime(p), print1(p, ", "))) \\ Iain Fox, Dec 19 2017

Extensions

Sequence corrected by Vincenzo Librandi, Jul 15 2012

A060844 Primes of the form 6*k^2 + 6*k + 31.

Original entry on oeis.org

31, 43, 67, 103, 151, 211, 283, 367, 463, 571, 691, 823, 967, 1123, 1291, 1471, 1663, 1867, 2083, 2311, 2551, 2803, 3067, 3343, 3631, 3931, 4243, 4567, 4903, 6367, 6763, 7591, 8467, 8923, 9391, 9871, 10867, 11383, 12451, 13003, 13567, 14143, 14731
Offset: 1

Views

Author

Tito Piezas III, May 03 2001

Keywords

Comments

Prime for n=[0,28]. Discriminant is -708, which is class no. 4.

References

  • Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 145.

Links

Crossrefs

Cf. A060834.

Programs

  • Mathematica
    Select[Table[6n^2+6n+31,{n,0,49}],PrimeQ] (* Stefano Spezia, Apr 17 2025 *)
  • PARI
    { n=0; for (m=0, 2136, f=6*m^2 + 6*m + 31; if (isprime(f), write("b060844.txt", n++, " ", f)); ) } \\ Harry J. Smith, Jul 13 2009

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), May 07 2001

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