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 10 results.

A103605 Pythagorean triples in increasing order of perimeter (a+b+c). If two successive perimeters are equals, then in order of decreasing areas; each triple [a(i), a(j), a(k)] (with k multiple of 3, j=k-1, i=k-2) in increasing order.

Original entry on oeis.org

3, 4, 5, 6, 8, 10, 5, 12, 13, 9, 12, 15, 8, 15, 17, 12, 16, 20, 7, 24, 25, 15, 20, 25, 10, 24, 26, 20, 21, 29, 18, 24, 30, 16, 30, 34, 21, 28, 35, 12, 35, 37, 15, 36, 39, 9, 40, 41, 24, 32, 40, 27, 36, 45, 14, 48, 50, 20, 48, 52, 24, 45, 51, 30, 40, 50, 28, 45, 53
Offset: 1

Views

Author

Alexandre Wajnberg, Mar 24 2005

Keywords

Comments

The corresponding perimeters A009096. - Wolfdieter Lang, Oct 06 2014

Links

Crossrefs

For primitive triples see A103606.
Cf. A009096, A376608.

Extensions

In the name second 'increasing' -> 'decreasing' (observed by A. Boe). - Wolfdieter Lang, Oct 06 2014

A136003 Primes that are not the sum, minus 1, of a Pythagorean triple.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 31, 37, 41, 43, 53, 61, 67, 73, 97, 101, 103, 109, 113, 127, 137, 151, 157, 163, 173, 193, 211, 229, 241, 257, 271, 277, 281, 283, 293, 313, 317, 331, 337, 353, 367, 397, 401, 409, 421, 433, 457, 463, 487, 499, 521, 523, 541, 547, 557
Offset: 1

Views

Author

Omar E. Pol, Dec 16 2007

Keywords

Comments

Primes in A136002.

Links

Crossrefs

Cf. A009096, A010814, A136002.

Programs

  • Mathematica
    q[n_] := PrimeQ[n] && (n == 2 || Module[{d = Divisors[(n+1)/2]}, AllTrue[Range[3, Length[d]], d[[#]] >= 2 * d[[#-1]] &]]); Select[Range[600], q] (* Amiram Eldar, Oct 19 2024 *)

Extensions

Extended by Ray Chandler, Dec 13 2008

A118905 Sum of legs of Pythagorean triangles (without multiple entries).

Original entry on oeis.org

7, 14, 17, 21, 23, 28, 31, 34, 35, 41, 42, 46, 47, 49, 51, 56, 62, 63, 68, 69, 70, 71, 73, 77, 79, 82, 84, 85, 89, 91, 92, 93, 94, 97, 98, 102, 103, 105, 112, 113, 115, 119, 123, 124, 126, 127, 133, 136, 137, 138, 140, 141, 142, 146, 147, 151, 153, 154, 155, 158, 161, 164, 167, 168, 170, 175, 178, 182, 184, 186, 187, 188
Offset: 1

Views

Author

Giovanni Resta, May 05 2006

Keywords

Comments

The prime numbers in this sequence define A001132 (see comment in A001132). - Richard Choulet, Dec 16 2008
For the sum of legs of Pythagorean triangles with multiple entries see A198390. - Wolfdieter Lang, May 24 2013
Are these just the positive multiples of A001132? - Charles R Greathouse IV, May 28 2013
For the sum of legs of primitive Pythagorean triangles see A120681. - Wolfdieter Lang, Feb 17 2015
n is in the sequence iff A331671(n) > 0. - Ray Chandler, Feb 26 2020

Examples

			7 = 3 + 4 and 3^2 + 4^2 = 5^2.
a(14) = 49 = 7^2 from the primitive Pythagorean triangle (x,y,z) = (9,40,41), and from the non-primitive one 7*(3,4,5); a(42) = 119 = 7*17 from four Pythagorean triangles (39,80,89) and (99,20,181) (both primitive) and 7*(5,12,13), 17*(3,4,5). - _Wolfdieter Lang_, May 24 2013

Links

Crossrefs

Cf. A009096, A118903, A118904, A058529, A001132, A120681, A331671.

Programs

  • Magma
    [m:m in [2..200]|#[k:k in [1..m-1]|IsSquare(k^2+(m-k)^2)] ne 0]; // Marius A. Burtea, Jul 29 2019
  • PARI
    is(n)=my(t=n^2); forstep(i=2-n%2, n-2, 2, if(issquare((t+i^2)/2), return(1))); 0 \\ Charles R Greathouse IV, May 28 2013

Extensions

More terms from 147 on. - Richard Choulet, Nov 24 2009
Name specified. - Wolfdieter Lang, May 24 2013

A136002 Numbers that are not the sum, minus 1, of a Pythagorean triple.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 84
Offset: 1

Views

Author

Omar E. Pol, Dec 15 2007

Keywords

Comments

Numbers that are not in A136000.

Links

Crossrefs

Cf. A009096, A010814, A136000.

Programs

  • Mathematica
    q[n_] := EvenQ[n] || Module[{d = Divisors[(n+1)/2]}, AllTrue[Range[3, Length[d]], d[[#]] >= 2 * d[[#-1]] &]]; Select[Range[100], q] (* Amiram Eldar, Oct 19 2024 *)

Extensions

Extended by Ray Chandler, Dec 13 2008

A099829 Smallest perimeter S such that at least n distinct Pythagorean triangles with this perimeter can be constructed.

Original entry on oeis.org

12, 60, 120, 240, 420, 720, 840, 840, 1680, 1680, 2520, 2520, 4620, 5040, 5040, 5040, 9240, 9240, 9240, 9240, 18480, 18480, 18480, 18480, 18480, 27720, 27720, 27720, 27720, 27720, 27720, 55440, 55440, 55440, 55440, 55440, 55440, 55440, 55440
Offset: 1

Views

Author

Hugo Pfoertner, Oct 27 2004

Keywords

Examples

			a(3)=120 because 120 is the smallest possible perimeter for which 3 different Pythgorean triangles exist: 120=20+48+52=24+45+51=30+40+50.

Links

Crossrefs

Cf. A099830 first perimeter with exact match of number of Pythagorean triangles, A009096 ordered perimeters of Pythagorean triangles.

Extensions

More terms from Ray Chandler, Oct 29 2004

A099830 Smallest perimeter S such that exactly n distinct Pythagorean triangles with this perimeter can be constructed.

Original entry on oeis.org

12, 60, 120, 240, 420, 720, 1320, 840, 2640, 1680, 3360, 2520, 4620, 7920, 7560, 5040, 10080, 17160, 10920, 9240, 40320, 25200, 28560, 21840, 18480, 60480, 41580, 46200, 36960, 32760, 27720, 78540, 60060, 129360, 134640, 115920, 85680, 65520, 83160
Offset: 1

Views

Author

Hugo Pfoertner, Oct 27 2004

Keywords

Comments

Least perimeter common to exactly n distinct Pythagorean triangles. - Lekraj Beedassy, Jun 07 2006

Examples

			a(7)=1320 because 1320 is the smallest possible perimeter for which exactly 7 different Pythgorean triangles exist: 1320 = 110+600+610 = 120+594+606 = 220+528+572 = 231+520+569 = 264+495+561 = 330+440+550 = 352+420+548.

Links

Crossrefs

Cf. A099829 first perimeter producing at least n Pythagorean triangles, A009096 ordered perimeters of Pythagorean triangles, A001399, A069905 partitions into 3 parts.

Extensions

More terms from Ray Chandler, Oct 29 2004

A136000 a(n) = A010814(n) - 1.

Original entry on oeis.org

11, 23, 29, 35, 39, 47, 55, 59, 69, 71, 79, 83, 89, 95, 107, 111, 119, 125, 131, 139, 143, 149, 153, 155, 159, 167, 175, 179, 181, 191, 197, 199, 203, 207, 209, 215, 219, 223, 227, 233, 239, 251, 259, 263, 269, 275, 279, 285, 287, 299, 305, 307, 311, 319, 323
Offset: 1

Views

Author

Omar E. Pol, Dec 10 2007

Keywords

Comments

Numbers of the form P-1 in increasing order, where P is the sum of a Pythagorean triple. Also P is the perimeter of a Pythagorean triangle. The open triangle represent a triangle instrument and, in general, any musical instrument. Positive integers are musician numbers or dancer number A136002.

Examples

			a(1) = 11 because {3,4,5} is a Pythagorean triple and 3+4+5 = 12 is the sum of a Pythagorean triple and 11+1 = 12, then we can write 3+4+5 = 11+1.

Links

Crossrefs

Cf. A010814, A136001, A136002, A009096 (perimeters of Pythagorean triangles).

Programs

  • Mathematica
    q[n_] := OddQ[n] && Module[{d = Divisors[(n+1)/2]}, AnyTrue[Range[3, Length[d]], d[[#]] < 2 * d[[#-1]] &]]; Select[Range[350], q] (* Amiram Eldar, Oct 19 2024 *)

Extensions

Definition corrected by R. J. Mathar, Dec 12 2007
Extended by Ray Chandler, Dec 13 2008

A118904 Perimeters of rectangles with integer sides and diagonal.

Original entry on oeis.org

14, 28, 34, 42, 46, 56, 62, 68, 70, 82, 84, 92, 94, 98, 102, 112, 124, 126, 136, 138, 140, 142, 146, 154, 158, 164, 168, 170, 178, 182, 184, 186, 188, 194, 196, 204, 206, 210, 224, 226, 230, 238, 246, 248, 252, 254, 266, 272, 274, 276, 280, 282, 284, 292, 294
Offset: 1

Views

Author

Giovanni Resta, May 05 2006

Keywords

Examples

			14 = 2*(3+4) and 3^2+4^2=5^2.

Links

Crossrefs

Cf. A009096, A118902, A118903, A118905.

Formula

Twice A118905.

A136001 Primes in A136000.

Original entry on oeis.org

11, 23, 29, 47, 59, 71, 79, 83, 89, 107, 131, 139, 149, 167, 179, 181, 191, 197, 199, 223, 227, 233, 239, 251, 263, 269, 307, 311, 347, 349, 359, 373, 379, 383, 389, 419, 431, 439, 443, 449, 461, 467, 479, 491, 503, 509, 563, 569, 571, 587, 593, 599, 607, 643
Offset: 1

Views

Author

Omar E. Pol, Dec 10 2007

Keywords

Examples

			a(1) = 11 because 11 is prime and {3,4,5} is a Pythagorean triple and 3+4+5 = 12 is the sum of a Pythagorean triple and 11+1 = 12, then we can write 3+4+5 = 11+1.

Links

Crossrefs

Cf. A136000, A136003, A009096 (perimeters of Pythagorean triangles).

Programs

  • Maple
    isprPer := proc(p) local dvs,m,n ; if p mod 2 = 1 then RETURN(false) ; fi ; dvs := p/2 ; for m in numtheory[divisors](dvs) do n := dvs/m-m ; if n > 0 and n < m then RETURN(true) ; fi ; od: RETURN(false) ; end: isA010814 := proc(n) local d; for d in numtheory[divisors](n) do if isprPer(n/d) then RETURN(true) ; fi ; od: RETURN(false) ; end: isA136000 := proc(n) isA010814(n+1) ; end: isA136001 := proc(n) isprime(n) and isA136000(n) ; end: for n from 2 to 600 do if isA136001(n) then printf("%d, ",n) ; fi: od: # R. J. Mathar, Dec 12 2007
  • Mathematica
    q[n_] := PrimeQ[n] && Module[{d = Divisors[(n+1)/2]}, AnyTrue[Range[3, Length[d]], d[[#]] < 2 * d[[#-1]] &]]; Select[Range[650], q] (* Amiram Eldar, Oct 19 2024 *)

Extensions

More terms from R. J. Mathar, Dec 12 2007
Extended by Ray Chandler, Dec 13 2008

A107487 Ordered semiperimeters of Pythagorean triangles.

Original entry on oeis.org

6, 12, 15, 18, 20, 24, 28, 30, 30, 35, 36, 40, 42, 42, 45, 45, 48, 54, 56, 60, 60, 60, 63, 66, 66, 70, 72, 72, 75, 77, 78, 80, 84, 84, 84, 88, 90, 90, 90, 91, 96, 99, 100, 102, 104, 105, 105, 108, 110, 112, 114, 117, 120, 120, 120, 120, 126, 126
Offset: 1

Views

Author

Lekraj Beedassy, May 28 2005

Keywords

Comments

For ordered semiperimeters of primitive Pythagorean triangles, see A020886.

Links

Crossrefs

Cf. A005279, A009096, A020886.

Formula

a(n) = A009096(n)/2.
Showing 1-10 of 10 results.

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