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A145010 a(n) = area of Pythagorean triangle with hypotenuse p, where p = A002144(n) = n-th prime == 1 mod 4. 3
6, 30, 60, 210, 210, 180, 630, 330, 1320, 1560, 2340, 990, 2730, 840, 4620, 3570, 5610, 4290, 1710, 7980, 2730, 6630, 10920, 12540, 4080, 8970, 14490, 18480, 9690, 3900, 11550, 25200, 26910, 30600, 34650, 32130, 37050, 7980, 23460, 6090, 29580, 49140, 35700 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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).

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

FORMULA

a(n) = A002365(n)*A002366(n)/2.

a(n) = x*y*(x^2-y^2), where x = A002330(n+1), y = A002331(n+1).

EXAMPLE

The following table shows the relationship between several closely related sequences:

Here p = A002144 = primes == 1 mod 4, p = a^2+b^2 with a < b;

a = A002331, b = A002330, t_1 = ab/2 = A070151;

p^2 = c^2+d^2 with c < d; c = A002366, d = A002365,

t_2 = 2ab = A145046, t_3 = b^2-a^2 = A070079,

with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).

  ---------------------------------

   p  a  b  t_1  c   d t_2 t_3  t_4

  ---------------------------------

   5  1  2   1   3   4   4   3    6

  13  2  3   3   5  12  12   5   30

  17  1  4   2   8  15   8  15   60

  29  2  5   5  20  21  20  21  210

  37  1  6   3  12  35  12  35  210

  41  4  5  10   9  40  40   9  180

  53  2  7   7  28  45  28  45  630

MATHEMATICA

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-Fran├žois Alcover, Feb 04 2015 *)

PROG

(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)", "))

(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

CROSSREFS

Cf. A002144, A002365, A002366, A144954.

Sequence in context: A065800 A181827 A263573 * A056835 A056836 A277521

Adjacent sequences:  A145007 A145008 A145009 * A145011 A145012 A145013

KEYWORD

nonn

AUTHOR

M. F. Hasler, Feb 24 2009

STATUS

approved

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Last modified October 18 18:10 EDT 2018. Contains 316323 sequences. (Running on oeis4.)