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A242219
Smallest a(n) in Pythagorean triple (a, b, c) such that c(n) - b(n) = n.
2
3, 4, 9, 8, 15, 12, 21, 12, 15, 20, 33, 24, 39, 28, 45, 24, 51, 24, 57, 40, 63, 44, 69, 36, 35, 52, 45, 56, 87, 60, 93, 40, 99, 68, 105, 48, 111, 76, 117, 60, 123, 84, 129, 88, 75, 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
OFFSET
1,1
COMMENTS
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.
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
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = min(sqrt(2n*h(i)-n^2), where h(i)=A009003(i)>n.
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
EXAMPLE
For n=7, a(7) = sqrt(2*7*h(12)-7^2) = 21;
for n=8, a(8) = sqrt(2*8*h(3)-8^2) = 12;
for n=9, a(9) = sqrt(2*9*h(5)-9^2) = 15;
for n=10, a(10) = sqrt(2*10*h(7)-10^2) = 20.
MATHEMATICA
(* first do *) hypos = A009003; (* then *)
Table[ First[ Union[ Flatten[ Table[ Select[{Sqrt[2 hypos[[i]]*n - n^2]}, IntegerQ && hypos[[i]] > n], {i, 1, Length[hypos]}]]]], {n, 1, 200}]
(* view table *) ListLinePlot[%]
PROG
(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
CROSSREFS
Sequence in context: A178590 A022463 A239384 * A106833 A217578 A280616
KEYWORD
nonn
AUTHOR
V.J. Pohjola, May 07 2014
STATUS
approved