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%I #19 Jan 15 2015 10:35:54
%S 3,5,15,21,35,9,45,11,55,39,65,99,91,15,105,51,85,165,19,95,195,221,
%T 105,209,255,69,115,231,285,25,75,175,299,225,275,189,325,399,391,29,
%U 145,351,425,261,459,279,341,165,231,575,465,551,35,105,609,315,589,385,675
%N a(n) gives the odd leg of the unique primitive Pythagorean triangle with hypotenuse A002144(n).
%C Consider sequence A002144 of primes congruent to 1 (mod 4) and equal to x^2 + y^2, with y>x given by A002330 and A002331; sequence gives values y^2 - x^2.
%C Odd legs of primitive Pythagorean triangles with unique (prime) hypotenuse (A002144), sorted on the latter. Corresponding even legs are given by 4*A070151 (or A145046). - _Lekraj Beedassy_, Jul 22 2005
%H T. D. Noe, <a href="/A070079/b070079.txt">Table of n, a(n) for n=1..1000</a>
%F a(n)=A079886(n)*A079887(n) - _Benoit Cloitre_, Jan 13 2003
%F a(n) is the odd positive integer with A080109(n) = A002144(n)^2 = a(n)^2 + (4*A070151(n))^2, in this unique decomposition into positive squares (up to order). See the _Lekraj Beedassy_, comment. - _Wolfdieter Lang_, Jan 13 2015
%e The following table shows the relationship
%e between several closely related sequences:
%e Here p = A002144 = primes == 1 mod 4, p = a^2+b^2 with a < b;
%e a = A002331, b = A002330, t_1 = ab/2 = A070151;
%e p^2 = c^2+d^2 with c < d; c = A002366, d = A002365,
%e t_2 = 2ab = A145046, t_3 = b^2-a^2 = A070079,
%e with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
%e ---------------------------------
%e .p..a..b..t_1..c...d.t_2.t_3..t_4
%e ---------------------------------
%e .5..1..2...1...3...4...4...3....6
%e 13..2..3...3...5..12..12...5...30
%e 17..1..4...2...8..15...8..15...60
%e 29..2..5...5..20..21..20..21..210
%e 37..1..6...3..12..35..12..35..210
%e 41..4..5..10...9..40..40...9..180
%e 53..2..7...7..28..45..28..45..630
%e .................................
%t pp = Select[ Range[200] // Prime, Mod[#, 4] == 1 &]; f[p_] := y^2 - x^2 /. ToRules[ Reduce[0 <= x <= y && p == x^2 + y^2, {x, y}, Integers]]; A070079 = f /@ pp (* _Jean-François Alcover_, Jan 15 2015 *)
%Y Cf. A002144, A002330, A002331, A080109, A070151
%K easy,nonn
%O 1,1
%A _Lekraj Beedassy_, May 06 2002
%E More terms from _Benoit Cloitre_, Jan 13 2003
%E Edited: Used a different name and moved old name to the comment section. - _Wolfdieter Lang_, Jan 13 2015
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