A144954 -id:A144954 - cp's OEIS frontend

A144960 Triangle of primes described in A144954, read by rows.

5, 37, 41, 157, 173, 181, 1277, 1289, 1297, 1301, 4441, 4457, 4481, 4493, 4513, 8669, 8677, 8681, 8689, 8693, 8713, 14533, 14537, 14549, 14557, 14561, 14593, 14621, 883241, 883249, 883273, 883357, 883397, 883409, 883429, 883433

Offset: 1

David Broadhurst, Feb 24 2009

Row 11 is a truncation of row 12.

			Triangle begins:
  [1, [5]],
  [2, [37, 41]],
  [3, [157, 173, 181]],
  [4, [1277, 1289, 1297, 1301]],
  [5, [4441, 4457, 4481, 4493, 4513]],
  [6, [8669, 8677, 8681, 8689, 8693, 8713]],
  [7, [14533, 14537, 14549, 14557, 14561, 14593, 14621]],
  [8, [883241, 883249, 883273, 883357, 883397, 883409, 883429, 883433]],
  [9, [10006957, 10006973, 10007021, 10007077, 10007161, 10007177, 10007201, 10007269, 10007273]],
  [10, [530551397, 530551457, 530551633, 530551669, 530551717, 530551733, 530551753, 530551757, 530551829, 530551841]],
  [11, [931953301, 931953389, 931953397, 931953409, 931953433, 931953437, 931953469, 931953509, 931953569, 931953637, 931953709]],
  [12, [931953301, 931953389, 931953397, 931953409, 931953433, 931953437, 931953469, 931953509, 931953569, 931953637, 931953709, 931953733]],
  ...

Cf. A144954.

A070151 a(n) is one fourth of the even leg of the unique primitive Pythagorean triangle with hypotenuse A002144(n).

Original entry on oeis.org

1, 3, 2, 5, 3, 10, 7, 15, 12, 20, 18, 5, 15, 28, 22, 35, 33, 13, 45, 42, 7, 15, 52, 30, 8, 65, 63, 40, 17, 78, 77, 72, 45, 68, 63, 85, 57, 10, 30, 105, 102, 70, 42, 95, 55, 110, 105, 133, 130, 12, 92, 60, 153, 152, 50, 143, 75, 138, 13, 65, 165, 27, 117, 190, 150, 187, 143, 70

Offset: 1

Lekraj Beedassy, May 06 2002

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 x*y/2.

			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
.................................
n = 7: a(7) = 7, A002144(7) = 53 and 53^2 = 2809 = A070079(7)^2 + (4*a(7))^2 = 45^2 + (4*7)^2 = 2025 + 784. - _Wolfdieter Lang_, Jan 13 2015

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

Cf. A002144, A002330, A002331, A070079, A080109, A144954, A144960.

a(n) = A002330(n+1)*A002331(n+1)/2. - David Wasserman, May 12 2003

4*a(n) is the even positive integer with A080109(n) = A002144(n)^2 = A070079(n)^2 + (4*a(n))^2 in this unique decomposition (up to order). See A080109 for references. - Wolfdieter Lang, Jan 13 2015

Edited. New name, moved the old one to the comment section. - Wolfdieter Lang, Jan 13 2015

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

Original entry on oeis.org

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

Offset: 1

M. F. Hasler, Feb 24 2009

nonn

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).

			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

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

Cf. A002144, A002365, A002366, A144954.

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 *)

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)","))

{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

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

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

cp's OEIS Frontend

A144960 Triangle of primes described in A144954, read by rows.

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A070151 a(n) is one fourth of the even leg of the unique primitive Pythagorean triangle with hypotenuse A002144(n).

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

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