cp's OEIS Frontend

A145010 a(n) = area of Pythagorean triangle with hypotenuse p, where p = A002144(n) = n-th prime == 1 (mod 4).

%I A145010 #19 Sep 05 2021 22:07:09
%S A145010 6,30,60,210,210,180,630,330,1320,1560,2340,990,2730,840,4620,3570,
%T A145010 5610,4290,1710,7980,2730,6630,10920,12540,4080,8970,14490,18480,9690,
%U A145010 3900,11550,25200,26910,30600,34650,32130,37050,7980,23460,6090,29580,49140,35700
%N A145010 a(n) = area of Pythagorean triangle with hypotenuse p, where p = A002144(n) = n-th prime == 1 (mod 4).
%C A145010 Pythagorean primes, i.e., primes of the form p = 4k+1 = A002144(n), have exactly one representation as sum of two squares: A002144(n) = x^2+y^2 = A002330(n+1)^2+A002331(n+1)^2. The corresponding (primitive) integer-sided right triangle with sides { 2xy, |x^2-y^2| } = { A002365(n), A002366(n) } has area xy|x^2-y^2| = a(n). For n>1 this is a(n) = 30*A068386(n).
%H A145010 T. D. Noe, <a href="/A145010/b145010.txt">Table of n, a(n) for n = 1..1000</a>
%F A145010 a(n) = A002365(n)*A002366(n)/2.
%F A145010 a(n) = x*y*(x^2-y^2), where x = A002330(n+1), y = A002331(n+1).
%e A145010 The following table shows the relationship between several closely related sequences:
%e A145010 Here p = A002144 = primes == 1 (mod 4), p = a^2+b^2 with a < b;
%e A145010 a = A002331, b = A002330, t_1 = ab/2 = A070151;
%e A145010 p^2 = c^2+d^2 with c < d; c = A002366, d = A002365,
%e A145010 t_2 = 2ab = A145046, t_3 = b^2-a^2 = A070079,
%e A145010 with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).
%e A145010   ---------------------------------
%e A145010    p  a  b  t_1  c   d t_2 t_3  t_4
%e A145010   ---------------------------------
%e A145010    5  1  2   1   3   4   4   3    6
%e A145010   13  2  3   3   5  12  12   5   30
%e A145010   17  1  4   2   8  15   8  15   60
%e A145010   29  2  5   5  20  21  20  21  210
%e A145010   37  1  6   3  12  35  12  35  210
%e A145010   41  4  5  10   9  40  40   9  180
%e A145010   53  2  7   7  28  45  28  45  630
%t A145010 Reap[For[p = 2, p < 500, p = NextPrime[p], If[Mod[p, 4] == 1, area = x*y/2 /. ToRules[Reduce[0 < x <= y && p^2 == x^2 + y^2, {x, y}, Integers]]; Sow[area]]]][[2, 1]] (* _Jean-François Alcover_, Feb 04 2015 *)
%o A145010 (PARI) forprime(p=1,499, p%4==1 | next; t=[p,lift(-sqrt(Mod(-1,p)))]; while(t[1]^2>p,t=[t[2],t[1]%t[2]]); print1(t[1]*t[2]*(t[1]^2-t[2]^2)","))
%o A145010 (PARI) {Q=Qfb(1,0,1);forprime(p=1,499,p%4==1|next;t=qfbsolve(Q,p); print1(t[1]*t[2]*(t[1]^2-t[2]^2)","))} \\ _David Broadhurst_
%Y A145010 Cf. A002144, A002365, A002366, A144954.
%K A145010 nonn
%O A145010 1,1
%A A145010 _M. F. Hasler_, Feb 24 2009