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A020886 Ordered semiperimeters of primitive Pythagorean triangles.

%I A020886 #45 Jan 01 2025 14:42:03
%S A020886 6,15,20,28,35,42,45,63,66,72,77,88,91,99,104,110,117,120,130,143,153,
%T A020886 156,165,170,187,190,195,204,209,210,221,228,231,238,247,255,266,272,
%U A020886 273,276,285,299,304,322,323,325,336,342,345,350,357,368,378,391,399
%N A020886 Ordered semiperimeters of primitive Pythagorean triangles.
%C A020886 k is in this sequence iff A078926(k) > 0.
%C A020886 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
%C A020886 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
%H A020886 Ray Chandler, <a href="/A020886/b020886.txt">Table of n, a(n) for n = 1..10000</a> (duplicates removed by Sean A. Irvine)
%F A020886 a(n) = A024364(n)/2.
%p A020886 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
%t A020886 A078926[n_] := Sum[Boole[n < d^2 < 2n && CoprimeQ[d, n/d]], {d, Divisors[ n/2^IntegerExponent[n, 2]]}];
%t A020886 Select[Range[1000], A078926[#]>0&] (* _Jean-François Alcover_, Mar 23 2020 *)
%o A020886 (PARI) 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(d2<nn && d2>n, return(1))); 0
%o A020886 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
%Y A020886 Subsequence of A005279.
%Y A020886 Cf. A024364, A078926, A020882, A020884, A020885.
%Y A020886 Triangles with 2/a = 1/b + 1/c:  A343891 (triples), A020883 (side a), A343892 (side b), A343893 (side c), A343894 (perimeter).
%K A020886 nonn
%O A020886 1,1
%A A020886 _Clark Kimberling_