A078926 -id:A078926 - cp's OEIS frontend

A078927 Smallest s for which there are exactly n primitive Pythagorean triangles with perimeter 2s; i.e., smallest s such that A078926(s) = n.

6, 858, 7140, 158730, 771342, 3120180, 9699690, 31651620, 119584290, 198843645, 229474245, 406816410, 281291010, 1412220810, 1673196525, 3457939485, 3234846615, 4360010655, 4573403835, 4127218095, 11532931410, 12929686770, 101268227775

Offset: 1

Dean Hickerson, Dec 15 2002

nonn

A Pythagorean triangle is a right triangle whose edge lengths are all integers; such a triangle is 'primitive' if the lengths are relatively prime.

			a(2)=858; the primitive Pythagorean triangles with edge lengths (364, 627, 725) and (195, 748, 773) both have perimeter 2*858 = 1716.

Derek J. C. Radden and Peter T. C. Radden, Table of n, a(n) for n=1..39 (terms 1 through 15 were computed by Derek J. C. Radden)

a(n) = A078928(n)/2. Cf. A078926.

oddpart[n_] := If[OddQ[n], n, oddpart[n/2]]; ct[p_] := Length[Select[Divisors[oddpart[p/2]], p/2<#^2

a(8) from Robert G. Wilson v, Dec 19 2002
a(9)-a(15) from Derek J C Radden, Dec 22 2012
a(16)-a(23) from Peter T. C. Radden, Dec 29 2012

A020886 Ordered semiperimeters of primitive Pythagorean triangles.

Original entry on oeis.org

6, 15, 20, 28, 35, 42, 45, 63, 66, 72, 77, 88, 91, 99, 104, 110, 117, 120, 130, 143, 153, 156, 165, 170, 187, 190, 195, 204, 209, 210, 221, 228, 231, 238, 247, 255, 266, 272, 273, 276, 285, 299, 304, 322, 323, 325, 336, 342, 345, 350, 357, 368, 378, 391, 399

Offset: 1

Clark Kimberling

nonn

k is in this sequence iff A078926(k) > 0.
Also, ordered sides c of primitive triples (a, b, c) for integer-sided triangles where side a is the harmonic mean of the 2 other sides b and c, i.e., 2/a = 1/b + 1/c with b < a < c (A343893). - Bernard Schott, May 06 2021
a(n) are the ordered radii of inscribed circles in squares, from which the tangents to the circles are cut off by primitive Pythagorean triangles. - Alexander M. Domashenko, Oct 17 2024

Ray Chandler, Table of n, a(n) for n = 1..10000 (duplicates removed by Sean A. Irvine)

Subsequence of A005279.
Cf. A024364, A078926, A020882, A020884, A020885.
Triangles with 2/a = 1/b + 1/c:  A343891 (triples), A020883 (side a), A343892 (side b), A343893 (side c), A343894 (perimeter).

isA020886 := proc(an) local r::integer,s::integer ; for r from floor((an/2)^(1/2)) to floor(an^(1/2)) do for s from r-1 to 1 by -2 do if r*(r+s) = an and gcd(r,s) < 2 then RETURN(true) ; fi ; if r*(r+s) < an then break ; fi ; od ; od : RETURN(false) ; end : for n from 2 to 400 do if isA020886(n) then printf("%d,",n) ; fi ; od ; # R. J. Mathar, Jun 08 2006

A078926[n_] := Sum[Boole[n < d^2 < 2n && CoprimeQ[d, n/d]], {d, Divisors[ n/2^IntegerExponent[n, 2]]}];
Select[Range[1000], A078926[#]>0&] (* Jean-François Alcover, Mar 23 2020 *)

is(n,f=factor(n))=my(P=apply(i->f[i,1]^f[i,2],[2-n%2..#f~]),nn=2*n); forvec(v=vector(#P,i,[0,1]), my(d=prod(i=1,#v,P[i]^v[i]),d2=d^2); if(d2n, return(1))); 0
list(lim)=my(v=List()); forfactored(n=6,lim\1, if(is(n[1],n[2]), listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Feb 03 2023

a(n) = A024364(n)/2.

A070109 Number of right integer triangles with perimeter n and relatively prime side lengths.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

Right integer triangles have integer areas: see A070142, A051516.

a(n) is nonzero iff n is in A024364.

			For n=30 there are A005044(30) = 19 integer triangles; only one is right: 5+12+13 = 30, 5^2+12^2 = 13^2; therefore a(30) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..20000 (obtained from the b-file of A078926)
Eric Weisstein's World of Mathematics, Right Triangle.
Eric Weisstein's World of Mathematics, Pythagorean Triples.
Reinhard Zumkeller, Integer-sided triangles

Cf. A070080, A070081, A070082, A051493, A070093, A070101, A070138, A070084, A070137.

unitaryDivisors[n_] := Cases[Divisors[n], d_ /; GCD[d, n/d] == 1];
A078926[n_] := Count[unitaryDivisors[n], d_ /; OddQ[d] && Sqrt[n] < d < Sqrt[2n]];
a[n_] := If[EvenQ[n], A078926[n/2], 0];
Table[a[n], {n, 1, 1716}] (* Jean-François Alcover, Oct 04 2021 *)

a(n) = A078926(n/2) if n is even; a(n)=0 if n is odd.
a(n) = A051493(n) - A070094(n) - A070102(n).
a(n) <= A024155(n).

Secondary offset added by Antti Karttunen, Oct 07 2017

A120089 Square perimeters of primitive Pythagorean triangles.

Original entry on oeis.org

144, 900, 3136, 8100, 17424, 23716, 33124, 43264, 54756, 57600, 93636, 115600, 139876, 144400, 166464, 174724, 207936, 213444, 244036, 298116, 304704, 357604, 414736, 422500, 476100, 490000, 541696, 571536, 640000, 722500, 746496, 756900

Offset: 1

Lekraj Beedassy, Jun 07 2006

nonn

Square entries of A024364.

Charlie Neder, Table of n, a(n) for n = 1..1000
P. Yiu, "Primitive Pythagorean triangles with square perimeters" in 'Recreational Mathematics' Chap. 6.2 pp. 50/360

Cf. A120090.

isA024364 := proc(an) local r::integer,s::integer ; for r from floor((an/4)^(1/2)) to floor((an/2)^(1/2)) do for s from r-1 to 1 by -2 do if 2*r*(r+s) = an and gcd(r,s) < 2 then RETURN(true) ; fi ; if 2*r*(r+s) < an then break ; fi ; od ; od : RETURN(false) ; end : isA120089 := proc(an) RETURN( issqr(an) and isA024364(an)) ; end: for n from 2 to 1200 do if isA120089(n^2) then printf("%d,",n^2) ; fi ; od ; # R. J. Mathar, Jun 08 2006

A078926[n_] := Sum[Boole[n < d^2 < 2n && CoprimeQ[d, n/d]], {d, Divisors[n/2^IntegerExponent[n, 2]]}];
Reap[For[k = 2, k <= 10^6, k += 2, If[A078926[k/2] > 0 && IntegerQ@Sqrt@k, Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Oct 25 2023 *)

a(n) = (2*u*v)^2, where u=sqrt(j/2) and v=sqrt(j+k) {for coprime pairs (j,k),j>k with odd k such that pairs (u,v),u

a(n) = A024364(k) = A000290(j) for some k and j. - R. J. Mathar, Jun 08 2006

Corrected and extended by R. J. Mathar, Jun 08 2006

A077177 Number of primitive Pythagorean triangles with perimeter equal to A002110(n), the product of the first n primes.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 3, 5, 8, 17, 34, 59, 111, 213, 396, 746, 1413, 2690, 5147, 9826, 18885, 36269, 69952, 134949, 260743, 504636, 978311, 1899832, 3692980, 7190329, 13994206, 27279898, 53195986

Offset: 1

Kermit Rose and Randall L Rathbun, Nov 29 2002

A Pythagorean triangle is a right triangle whose edge lengths are all integers; such a triangle is 'primitive' if the lengths are relatively prime.

Equivalently, number of divisors of s=A070826(n) in the range (sqrt(s), sqrt(2s)). More generally, for any positive integer s, the number of primitive Pythagorean triangles with perimeter 2's equals the number of odd unitary divisors of s in the range (sqrt(s), sqrt(2s)). (A divisor d of n is 'unitary' if gcd(d, n/d) = 1.)

			a(5) = 1 since there is exactly one primitive Pythagorean triangle with perimeter 2*3*5*7*11; its edge lengths are (132, 1085, 1093). a(7) = 3; the 3 triangles have edge lengths (70941, 214060, 225509), (96460, 195789, 218261) and (142428, 156485, 211597).

A. S. Anema, "Pythagorean Triangles with Equal Perimeters", Scripta Mathematica, vol. 15 (1949) p. 89.
Albert H. Beiler, "Recreations in the Theory of Numbers", chapter XIV, "The Eternal Triangle", pp. 131, 132.
F. L. Miksa, "Pythagorean Triangles with Equal Perimeters", Mathematics, vol. 24 (1950), p. 52.

Randall L. Rathbun, Equal Perimeter primitive right triangles

Cf. A002110, A070109, A070826, A078926.

a[n_] := Length[Select[Divisors[s=Times@@Prime/@Range[2, n]], s<#^2<2s&]]

semi_peri(p)= {local(q,r,ct,tot); ct=0; tot=0; pt=0; fordiv(p,q,r=p/q-q; if(r<=q&&r>0,print(q,",",r," [",gcd(q,r),"] "); if(gcd(q,r)==1,ct=ct+1; if(q*r%2==0,pt=pt+1; ); ); tot=tot+1); ); print("semiperimeter:"p," Total sets:",tot," Coprime:",ct," Primitive:",pt); } /* Lists all pairs q,r such that the triangle with edge lengths (q^2-r^2, 2qr, q^2+r^2) has semiperimeter p. */

a(n) = A070109(A002110(n)) = A078926(A070826(n)).

Edited by Dean Hickerson, Dec 18 2002

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A078927 Smallest s for which there are exactly n primitive Pythagorean triangles with perimeter 2s; i.e., smallest s such that A078926(s) = n.

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A020886 Ordered semiperimeters of primitive Pythagorean triangles.

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A070109 Number of right integer triangles with perimeter n and relatively prime side lengths.

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A120089 Square perimeters of primitive Pythagorean triangles.

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A077177 Number of primitive Pythagorean triangles with perimeter equal to A002110(n), the product of the first n primes.

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