A081859
Short leg of primitive Pythagorean triangles sorted on semiperimeter.
Original entry on oeis.org
3, 5, 8, 7, 20, 12, 9, 28, 11, 16, 33, 48, 13, 36, 39, 20, 65, 15, 60, 44, 17, 24, 88, 51, 85, 19, 52, 119, 57, 28, 104, 95, 21, 84, 133, 60, 140, 32, 105, 23, 120, 69, 96, 115, 68, 25, 160, 36, 161, 75, 136, 207, 27, 204, 76, 175, 180, 40, 225, 135, 29, 152, 252, 189, 120
Offset: 1
A120210
Integer squares y from the smallest solutions of y^2 = x*(a^N - x)*(b^N + x) (elliptic line, Weierstrass equation) with a and b legs in primitive Pythagorean triangles and N = 2. Sequence ordered in increasing values of leg a.
Original entry on oeis.org
20, 30, 156, 600, 420, 1640, 3660, 520, 2590, 7140, 1224, 10920, 8190, 20880, 32580, 4872, 19998, 5220, 48620, 69960, 3150, 41470, 97656, 132860, 19080, 76830, 176820, 230880, 131070, 12740, 296480, 11100, 375156, 52360, 209950, 468540, 64080
Offset: 1
First primitive Pythagorean triple: 3, 4, 5.
Weierstrass equation: y^2 = x*(3^2 - x)*(4^2 + x).
Smallest integer solution: (x, y) = (4,20).
First element in the sequence: y = 20.
- G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 47.
A155180
Short leg A of primitive Pythagorean triangles such that perimeters and products of 3 sides are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2*p*q, ar=a*b/2; s=a+b+c, s-+1 are primes, pr=a*b*c, pr-+1 are primes.
Original entry on oeis.org
3, 15833, 71765, 75633, 94983, 256859, 263661, 292943, 309599, 315159, 340439, 349929, 375089, 415659, 416079, 445775, 446285, 525005, 583089, 639651, 655205, 663255, 707715, 953363, 955319, 988415, 1044051, 1074909, 1081365, 1116323
Offset: 1
Cf.
A020882,
A020886,
A020884,
A020883,
A024364,
A024406,
A155171,
A155173,
A155174,
A155175,
A155176,
A155177,
A155178
-
lst={};Do[p=n;q=p+1;a=q^2-p^2;c=q^2+p^2;b=2*p*q;ar=a*b/2;s=a+b+c;pr=a*b*c;If[PrimeQ[s-1]&&PrimeQ[s+1]&&PrimeQ[pr-1]&&PrimeQ[pr+1],AppendTo[lst,a]],{n,3*9!}];lst
A376428
Numbers k that occur as shorter legs of Pythagorean triangles with Pythagorean primes A002144 as hypotenuses.
Original entry on oeis.org
3, 5, 8, 9, 11, 12, 15, 19, 20, 25, 28, 29, 32, 35, 39, 40, 45, 48, 49, 51, 52, 59, 60, 61, 65, 68, 69, 71, 72, 75, 79, 80, 85, 88, 95, 101, 105, 108, 112, 115, 120, 121, 129, 131, 132, 139, 140, 141, 145, 148, 159, 160, 165, 168, 169, 171, 175, 180, 181, 188, 189
Offset: 1
Original entry on oeis.org
5, 13, 113, 1741, 5101, 8581, 9941, 21841, 26681, 47741, 82013, 481181, 501001, 1009621, 2356621, 2542513, 3279361, 3723721, 4277813, 7757861, 8124481, 13204661, 25311613, 30772013, 44170601, 48619661, 51521401, 52541501, 54236113, 60731221, 72902813
Offset: 1
Cf.
A020882,
A020886,
A020884,
A020883,
A024364,
A024406,
A155171,
A155173,
A155174,
A155175,
A155176,
A155177,
A155178,
A155180,
A088483,
A001844,
A096891,
A066885,
A099776,
A110494,
A081589
-
lst={};Do[p=n;q=p+1;a=q^2-p^2;c=q^2+p^2;b=2*p*q;ar=a*b/2;s=a+b+c;If[PrimeQ[s-1]&&PrimeQ[s+1],If[PrimeQ[c],AppendTo[lst,c]]],{n,8!}];lst (* corrected by Ray Chandler, Feb 11 2020 *)
A165260
Short legs of primitive Pythagorean triples which have a perimeter which is the average of a twin prime pair.
Original entry on oeis.org
3, 5, 15, 21, 24, 28, 36, 41, 59, 64, 89, 100, 101, 120, 131, 132, 141, 153, 155, 168, 180, 203, 204, 208, 209, 215, 220, 231, 244, 280, 288, 300, 309, 315, 336, 341, 348, 351, 395, 405, 408, 429, 448, 453, 455, 495, 520, 540, 551, 567, 568, 580, 592, 636, 648
Offset: 1
Triples (a,b,c) which satisfy the rules are (3,4,5), (5,12,13), (15,112,113), (21,220,221), (24,143,145), (28,195,197), (36,77,85), (41,840,841), (59,1740,1741), (64,1023,1025), (89,3960,3961), (100,2499,2501), ... 3+4+5=12 -> 11 and 13 are primes, 5+12+13=30 -> 29 and 31 are primes, ...
-
isA014574 := proc(n)
return ( isprime(n-1) and isprime(n+1) ) ;
end proc:
isA165260 := proc(n)
local d,bplc,b,c ;
for d in numtheory[divisors](n^2) do
bplc := n^2/d ;
c := (d+bplc)/2 ;
b := (bplc-d)/2 ;
if type(c,'integer') and type(b,'integer') then
if c > b and b >= n then
if igcd(n,b,c) = 1 and isA014574(n+b+c) then
return true;
end if;
end if;
end if;
end do:
return false;
end proc:
for n from 3 to 600 do
if isA165260(n) then
printf("%d,",n);
end if;
end do: # R. J. Mathar, Oct 29 2011
-
amax=10^4;lst={};k=0;q=12!;Do[If[(e=((n+1)^2-n^2))>amax,Break[]];Do[If[GCD[m,n]==1,a=m^2-n^2;b=2*m*n;If[GCD[a,b]==1,If[a>b,{a,b}={b,a}];If[a>amax,Break[]];c=m^2+n^2;x=a+b+c;If[PrimeQ[x-1]&&PrimeQ[x+1],k++;AppendTo[lst,a]]]],{m,n+1,12!,2}],{n,1,q,1}];Union@lst
A239581
Number of primitive Pythagorean triangles (x, y, z) with legs x < y < 10^n.
Original entry on oeis.org
1, 18, 179, 1788, 17861, 178600, 1786011, 17860355, 178603639, 1786036410, 17860362941
Offset: 1
a(1) = 1, because the only primitive Pythagorean triangle with x < y < 10 is [3, 4, 5].
A239744
Number of Pythagorean triangles (x, y, z) with legs x < y <= 10^n.
Original entry on oeis.org
2, 63, 1034, 14474, 185864, 2269788, 26809924, 309224756, 3503496007, 39147452729, 432599522197
Offset: 1
a(1) = 2, because the only two Pythagorean triangles with x < y < 10 are [3, 4, 5] and [6, 8, 10].
A239786
Number of Pythagorean triangles (x, y, z) with legs x < y < 10^n.
Original entry on oeis.org
2, 62, 1032, 14471, 185860, 2269783, 26809918, 309224749, 3503495999, 39147452720, 432599522187
Offset: 1
A138043
Concatenation of primitive Pythagorean triangles, sorted.
Original entry on oeis.org
345, 51213, 72425, 81517, 94041, 116061, 123537, 166365, 202129, 284553, 335665, 367785, 398089, 485573, 657297, 2099101, 6091109, 15112113, 17144145, 19180181, 21220221, 23264265, 24143145, 25312313, 28195197, 29420421, 31480481
Offset: 1
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