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A165260
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Short legs of primitive Pythagorean triples which have a perimeter which is the average of a twin prime pair.
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2
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3, 5, 15, 21, 24, 28, 36, 41, 59, 64, 89, 100, 101, 120, 131, 132, 141, 153, 155, 168, 180, 203, 204, 208, 209, 215, 220, 231, 244, 280, 288, 300, 309, 315, 336, 341, 348, 351, 395, 405, 408, 429, 448, 453, 455, 495, 520, 540, 551, 567, 568, 580, 592, 636, 648
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OFFSET
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1,1
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LINKS
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Table of n, a(n) for n=1..55.
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EXAMPLE
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Triples (a,b,c) which satisfy the rules are (3,4,5), (5,12,13), (15,112,113), (21,220,221), (24,143,145), (28,195,197), (36,77,85), (41,840,841), (59,1740,1741), (64,1023,1025), (89,3960,3961), (100,2499,2501), ... 3+4+5=12 -> 11 and 13 are primes, 5+12+13=30 -> 29 and 31 are primes, ...
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MAPLE
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isA014574 := proc(n)
return ( isprime(n-1) and isprime(n+1) ) ;
end proc:
isA165260 := proc(n)
local d, bplc, b, c ;
for d in numtheory[divisors](n^2) do
bplc := n^2/d ;
c := (d+bplc)/2 ;
b := (bplc-d)/2 ;
if type(c, 'integer') and type(b, 'integer') then
if c > b and b >= n then
if igcd(n, b, c) = 1 and isA014574(n+b+c) then
return true;
end if;
end if;
end if;
end do:
return false;
end proc:
for n from 3 to 600 do
if isA165260(n) then
printf("%d, ", n);
end if;
end do: # R. J. Mathar, Oct 29 2011
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MATHEMATICA
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amax=10^4; lst={}; k=0; q=12!; Do[If[(e=((n+1)^2-n^2))>amax, Break[]]; Do[If[GCD[m, n]==1, a=m^2-n^2; b=2*m*n; If[GCD[a, b]==1, If[a>b, {a, b}={b, a}]; If[a>amax, Break[]]; c=m^2+n^2; x=a+b+c; If[PrimeQ[x-1]&&PrimeQ[x+1], k++; AppendTo[lst, a]]]], {m, n+1, 12!, 2}], {n, 1, q, 1}]; Union@lst
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CROSSREFS
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Cf. A020884, A014574, A009004, A165261, A165262.
Sequence in context: A344991 A063185 A208854 * A201874 A059528 A070079
Adjacent sequences: A165257 A165258 A165259 * A165261 A165262 A165263
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KEYWORD
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nonn
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AUTHOR
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Vladimir Joseph Stephan Orlovsky, Sep 11 2009
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STATUS
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approved
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