# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a070079 Showing 1-1 of 1 %I A070079 #19 Jan 15 2015 10:35:54 %S A070079 3,5,15,21,35,9,45,11,55,39,65,99,91,15,105,51,85,165,19,95,195,221, %T A070079 105,209,255,69,115,231,285,25,75,175,299,225,275,189,325,399,391,29, %U A070079 145,351,425,261,459,279,341,165,231,575,465,551,35,105,609,315,589,385,675 %N A070079 a(n) gives the odd leg of the unique primitive Pythagorean triangle with hypotenuse A002144(n). %C A070079 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 A070079 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 A070079 T. D. Noe, Table of n, a(n) for n=1..1000 %F A070079 a(n)=A079886(n)*A079887(n) - _Benoit Cloitre_, Jan 13 2003 %F A070079 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 A070079 The following table shows the relationship %e A070079 between several closely related sequences: %e A070079 Here p = A002144 = primes == 1 mod 4, p = a^2+b^2 with a < b; %e A070079 a = A002331, b = A002330, t_1 = ab/2 = A070151; %e A070079 p^2 = c^2+d^2 with c < d; c = A002366, d = A002365, %e A070079 t_2 = 2ab = A145046, t_3 = b^2-a^2 = A070079, %e A070079 with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2). %e A070079 --------------------------------- %e A070079 .p..a..b..t_1..c...d.t_2.t_3..t_4 %e A070079 --------------------------------- %e A070079 .5..1..2...1...3...4...4...3....6 %e A070079 13..2..3...3...5..12..12...5...30 %e A070079 17..1..4...2...8..15...8..15...60 %e A070079 29..2..5...5..20..21..20..21..210 %e A070079 37..1..6...3..12..35..12..35..210 %e A070079 41..4..5..10...9..40..40...9..180 %e A070079 53..2..7...7..28..45..28..45..630 %e A070079 ................................. %t A070079 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 A070079 Cf. A002144, A002330, A002331, A080109, A070151 %K A070079 easy,nonn %O A070079 1,1 %A A070079 _Lekraj Beedassy_, May 06 2002 %E A070079 More terms from _Benoit Cloitre_, Jan 13 2003 %E A070079 Edited: Used a different name and moved old name to the comment section. - _Wolfdieter Lang_, Jan 13 2015 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE