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A242219 Smallest a(n) in Pythagorean triple (a, b, c) such that c(n) - b(n) = n.

%I A242219 #23 May 17 2014 11:35:03
%S A242219 3,4,9,8,15,12,21,12,15,20,33,24,39,28,45,24,51,24,57,40,63,44,69,36,
%T A242219 35,52,45,56,87,60,93,40,99,68,105,48,111,76,117,60,123,84,129,88,75,
%U A242219 92,141,72,63,60,153,104,159,72,165,84,171,116,177,120,183,124,105,80,195,132,201,136,207,140,213,84,219,148,105,152,231,156
%N A242219 Smallest a(n) in Pythagorean triple (a, b, c) such that c(n) - b(n) = n.
%C A242219 The local minima a(n) predominantly fluctuate, with an increasing amplitude, between the multiples of the leg lengths of the smallest primitive triple (3,4,5) and of its symmetric counterpart (4,3,5). When n grows, minima appear from higher primitive triples which further increase the amplitude.
%C A242219 We have a^2 = c^2 - b^2 = (c-b)(c+b) = n*(c+b). To find the least such square, use n=core(n)*f^2 with core = A007913, f = A000188(n), and look for the least c+b = c-b+2b = n+2b = core(n)*x^2 or x^2 = (n+2b)/core(n) = f^2 + 2b/core(n). The least such integer x is f+1 if core(n) is even, or else f+2. - _M. F. Hasler_, May 08 2014
%H A242219 Charles R Greathouse IV, <a href="/A242219/b242219.txt">Table of n, a(n) for n = 1..10000</a>
%F A242219 a(n) = min(sqrt(2n*h(i)-n^2), where h(i)=A009003(i)>n.
%F A242219 a(n) = sqrt(( n + (x^2 - f^2)*core(n))*n ) where f = A000188(n), x = f+1 if core(n) = A007913(n) = n / f^2 is even, x = f+2 if core(n) is odd. - _M. F. Hasler_, May 08 2014
%e A242219 For n=7, a(7) = sqrt(2*7*h(12)-7^2) = 21;
%e A242219 for n=8, a(8) = sqrt(2*8*h(3)-8^2) = 12;
%e A242219 for n=9, a(9) = sqrt(2*9*h(5)-9^2) = 15;
%e A242219 for n=10, a(10) = sqrt(2*10*h(7)-10^2) = 20.
%t A242219 (* first do *) hypos = A009003; (* then *)
%t A242219 Table[ First[ Union[ Flatten[ Table[ Select[{Sqrt[2 hypos[[i]]*n - n^2]}, IntegerQ && hypos[[i]] > n], {i, 1, Length[hypos]}]]]], {n, 1, 200}]
%t A242219 (* view table *) ListLinePlot[%]
%o A242219 (PARI)  a(n)={ my( f=core(n,1)); sqrtint(( if( bittest( f[1],0), 4*f[2]+4, 2*f[2]+1)*f[1]+n )*n )} \\ _M. F. Hasler_, May 08 2014
%K A242219 nonn
%O A242219 1,1
%A A242219 _V.J. Pohjola_, May 07 2014