A020884 -id:A020884 - cp's OEIS frontend

A155176 Perimeter s/6 (divided by 6) of primitive Pythagorean triangles such that perimeters are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2pq, s=a+b+c, s-+1 are primes.

2, 5, 40, 77, 287, 590, 1335, 1717, 2882, 3337, 3927, 4030, 6902, 7315, 7740, 8932, 15965, 20592, 26070, 27405, 34277, 34580, 40920, 50692, 92132, 96647, 113575, 139690, 160557, 167167, 220225, 237407, 279720, 300832, 310765, 336777, 389895

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

p=1,q=2,a=3,b=4,c=5,s=12-+1primes, ...

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173, A155174, A155175

lst={};Do[p=n;q=p+1;a=q^2-p^2;c=q^2+p^2;b=2*p*q;s=a+b+c;If[PrimeQ[s-1]&&PrimeQ[s+1],AppendTo[lst,s/6]],{n,8!}];lst

A009188 Short leg of more than one Pythagorean triangle.

Original entry on oeis.org

9, 12, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 49, 50, 51, 52, 54, 55, 56, 57, 60, 63, 64, 65, 66, 68, 69, 70, 72, 75, 76, 77, 78, 80, 81, 84, 85, 87, 88, 90, 91, 92, 93, 95, 96, 98, 99, 100, 102, 104, 105, 108, 110, 111, 112, 114, 115, 116

Offset: 1

David W. Wilson

nonn

Values of n for which composite n X n magic squares are possible. - J. Lowell, May 20 2010
If n is in the sequence, k*n is in the sequence for all k > 1. So odd semiprimes (A046315) and numbers of the form 4*p where p is an odd prime are core subsequences which give the initial terms of arithmetic progressions in this sequence. - Altug Alkan, Nov 29 2015
Numbers appearing more than once in A009004. - Sean A. Irvine, Apr 20 2018

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001749, A009004, A020884, A046315, A264828.

filter:= proc(n) not isprime(n) and (n::odd or not isprime(n/2)) end proc:
select(filter, [$9 .. 10000]); # Robert Israel, Nov 30 2015

filterQ[n_] := !PrimeQ[n] && (OddQ[n] || !PrimeQ[n/2]);
Select[Range[9, 120], filterQ] (* Jean-François Alcover, Feb 28 2019, from Maple *)

forcomposite(n=9, 1e3, if(n % 2 == 1 || !isprime(n/2), print1(n, ", "))) \\ Altug Alkan, Dec 01 2015

from sympy import primepi
def A009188(n):
    def f(x): return int(n+2+primepi(x)+primepi(x>>1))
    m, k = n+2, f(n+2)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Oct 17 2024

a(n) = A264828(n+2). - Chai Wah Wu, Oct 17 2024

A155177 Area ar/6 (divided by 6) of primitive Pythagorean triangles such that perimeters are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2pq, ar=a*b/2; s=a+b+c, s-+1 are primes.

Original entry on oeis.org

1, 5, 140, 385, 2870, 8555, 29370, 42925, 93665, 116795, 149226, 155155, 348551, 380380, 414090, 513590, 1229305, 1801800, 2567895, 2767905, 3873301, 3924830, 5053620, 6970150, 17090486, 18362930, 23396450, 31919165, 39336465, 41791750

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

p=1,q=2,a=3,b=4,c=5, ar=3*4/2=6, s=12-+1primes, ...

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173, A155174, A155175, A155176

lst={};Do[p=n;q=p+1;a=q^2-p^2;c=q^2+p^2;b=2*p*q;s=a+b+c;ar=a*b/2;If[PrimeQ[s-1]&&PrimeQ[s+1],AppendTo[lst,ar/6]],{n,8!}];lst

A381005 Ordered short legs of the Pythagorean triangles defined by a = 2^(4n) + 2^(2n+1), b = 2^(4n) - 2^(4n-2) - 2^(2n) - 1, c = 2^(4n) + 2^(4n-2) + 2^(2n) + 1.

Original entry on oeis.org

7, 175, 3007, 48895, 785407, 12578815, 201310207, 3221159935, 51539345407, 824632672255, 13194135339007, 211106215755775, 3377699653419007, 54043195260010495, 864691127381393407, 13835058050987196415, 221360928867334750207, 3541774862083514433535, 56668397794160864657407

Offset: 1

Robert C. Lyons, Feb 12 2025

Proper subset of A020884.
Conjecture: These Pythagorean triangles are primitive. Verified up to n=100000.
The preceding conjecture is true, since, for n>=1, the values of a,b,c are given by Euclid's formula for generating Pythagorean triples: a=2xy, b=x^2-y^2, c=x^2+y^2 with x=2^(2n) and y=2^(2n-1)+1 and x and y are coprime and x is even and y is odd. - Chai Wah Wu, Feb 13 2025

Paolo Xausa, Table of n, a(n) for n = 1..800
John D. Cook, Sparse binary Pythagorean triples (2025).
H. S. Uhler, A Colossal Primitive Pythagorean Triangle, The American Mathematical Monthly, Vol. 57, No. 5 (May, 1950), pp. 331-332.
Wikipedia, Pythagorean triple.
Index entries for linear recurrences with constant coefficients, signature (21,-84,64).

Cf. A020884.

Cf. A381006 (long legs), A381007 (hypotenuses), A381008 (perimeters), A381009 (areas).

[2^(4*n) - 2^(4*n-2) - 2^(2*n) - 1: n in [1..20]];

A381005[n_] := (3*# + 2)*(# - 2)/4 & [4^n]; Array[A381005, 20] (* or *)
LinearRecurrence[{21, -84, 64}, {7, 175, 3007}, 20] (* Paolo Xausa, Feb 26 2025 *)

a(n) = 2^(4*n) - 2^(4*n-2) - 2^(2*n) - 1

def A381005(n): return ((m:=1<<(n<<1)-1)-1)*(3*m+1) # Chai Wah Wu, Feb 13 2025

a(n) = 2^(4n) - 2^(4n-2) - 2^(2n) - 1.
a(n) = sqrt( A381007(n)^2 - A381006(n)^2 ).
G.f.: (7 + 28*x - 80*x^2)/((1 - x)*(1 - 4*x)*(1 - 16*x)). - Stefano Spezia, Feb 13 2025

A081925 Ordered even short legs of primitive Pythagorean triangles.

Original entry on oeis.org

8, 12, 16, 20, 20, 24, 28, 28, 32, 36, 36, 40, 44, 44, 48, 48, 52, 52, 56, 60, 60, 60, 64, 68, 68, 72, 76, 76, 80, 84, 84, 84, 88, 88, 92, 92, 96, 96, 100, 100, 104, 104, 108, 108, 112, 116, 116, 120, 120, 120, 124, 124, 128, 132, 132, 132, 136, 136, 140, 140, 140, 144

Offset: 1

Lekraj Beedassy, Apr 23 2003

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A020884.

Corrected and extended by Ray Chandler, Oct 29 2003

Offset corrected by Michel Marcus, Nov 12 2019

A120211 x values giving the smallest integer solutions of y^2 = x(a^N - x)( b^N + x) (elliptic curve, Weierstrass equation) with a and b legs in primitive Pythagorean triangles and N = 2. Sequence ordered in increasing values of leg a. Relevant y values in A120210.

Original entry on oeis.org

4, 6, 12, 24, 15, 40, 60, 40, 70, 84, 72, 56, 126, 144, 180, 168, 198, 180, 220, 264, 126, 286, 312, 364, 360, 390, 420, 480, 510, 49, 544, 300, 612, 616, 646, 684, 720, 760, 288, 798, 840, 924, 726, 966, 700, 1012, 1104, 990, 1150, 1200

Offset: 1

Giorgio Balzarotti, Paolo P. Lava, Jun 10 2006

nonn

			First primitive Pythagorean triad: 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 x = 4

G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 47.

Cf. A009003, A020884, A120210-A120213.

flag :=1;x:=0; # a, b, c primitive Pythagorean triad while flag =1 do x:=x+1; y2:= x*( a^2 - x)*(x+b^2); if ((floor(sqrt(y2)))^2=y2)then print( x);flag :=0;fi; od;

A155178 Numbers p 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=2pq, ar=ab/2; s=a+b+c, s-+1 are primes, pr=ab*c, pr-+1 are primes.

Original entry on oeis.org

1, 7916, 35882, 37816, 47491, 128429, 131830, 146471, 154799, 157579, 170219, 174964, 187544, 207829, 208039, 222887, 223142, 262502, 291544, 319825, 327602, 331627, 353857, 476681, 477659, 494207, 522025, 537454, 540682, 558161, 571670

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

p=1,q=2,a=3,b=4,c=5,s=12-+1 primes,pr=3*4*5=60-+1 primes, ...

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173, A155174, A155175, A155176, A155177

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,n]],{n,3*9!}];lst

A024355 Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence gives values of AUB, sorted.

Original entry on oeis.org

3, 4, 5, 7, 8, 9, 11, 12, 12, 13, 15, 15, 16, 17, 19, 20, 20, 21, 21, 23, 24, 24, 25, 27, 28, 28, 29, 31, 32, 33, 33, 35, 35, 36, 36, 37, 39, 39, 40, 40, 41, 43, 44, 44, 45, 45, 47, 48, 48, 49, 51, 51, 52, 52, 53, 55, 55, 56, 56, 57, 57, 59, 60, 60, 60, 60, 61, 63, 63, 64, 65

Offset: 1

David W. Wilson

nonn

Union of A020883 and A020884, sorted (with multiplicity); n occurs A024361(n) times. - Ray Chandler, Feb 03 2020

Ray Chandler, Table of n, a(n) for n = 1..10000
Ron Knott, Pythagorean Triples and Online Calculators

Cf. A020883, A020884, A024361.

A024357 Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence gives values of AUBUC, sorted.

Original entry on oeis.org

3, 4, 5, 5, 7, 8, 9, 11, 12, 12, 13, 13, 15, 15, 16, 17, 17, 19, 20, 20, 21, 21, 23, 24, 24, 25, 25, 27, 28, 28, 29, 29, 31, 32, 33, 33, 35, 35, 36, 36, 37, 37, 39, 39, 40, 40, 41, 41, 43, 44, 44, 45, 45, 47, 48, 48, 49, 51, 51, 52, 52, 53, 53, 55, 55, 56, 56, 57

Offset: 1

David W. Wilson

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000
Ron Knott, Pythagorean Triples and Online Calculators

Cf. A020882, A020883, A020884, A042965.

A024411 Short leg of more than one primitive Pythagorean triangle.

Original entry on oeis.org

20, 28, 33, 36, 39, 44, 48, 51, 52, 57, 60, 65, 68, 69, 75, 76, 84, 85, 87, 88, 92, 93, 95, 96, 100, 104, 105, 108, 111, 115, 116, 119, 120, 123, 124, 129, 132, 133, 135, 136, 140, 141, 145, 147, 148, 152, 155, 156, 159, 160, 161, 164, 165, 168, 172, 175, 177, 180, 183, 184

Offset: 1

David W. Wilson

nonn

Every term is composite. - Clark Kimberling, Feb 04 2024

Proof by contradiction: let p prime be the short leg. Then p^2 + b^2 = c^2 i.e., p^2 = (c - b) * (c + b). Then (c - b, c + b) in {(1, p^2), (p, p)}. If (c - b, c + b) = (p, p) then c = p and b = 0 which is impossible. Hence there is at most one solution for (c - b, c + b). A contradiction. - David A. Corneth, Feb 04 2024

Ray Chandler, Table of n, a(n) for n = 1..10000
Ron Knott, Pythagorean Triples and Online Calculators

Cf. A020884, A024352.

aa=1;s="";For[a=1,a<=10^2,For[b=a+1,((b+1)^2-b^2)<=a^2,c=(a^2+b^2)^0.5;If[c==Round[c]&&GCD[a,b]==1,If[a==aa,s=s<>ToString[a]<>","];If[a!=aa,aa=a,aa=1]];b++ ];a++ ];s (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)

is(n) = {
	my(d = divisors(n^2), q = 0, b, c);
	for(i = 1, #d\2,
		if(!bitand(d[#d + 1 - i] - d[i], 1),
			c = (d[i] + d[#d + 1 - i])/2;
			b = d[#d + 1 - i] - c;
			if(gcd(n, b) == 1 && n < b,
				q++;
				if(q >= 2,
					return(1)
				)
			)
		)
	); 0
} \\ David A. Corneth, Feb 04 2024

cp's OEIS Frontend

A155176 Perimeter s/6 (divided by 6) of primitive Pythagorean triangles such that perimeters are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2*p*q, s=a+b+c, s-+1 are primes.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A009188 Short leg of more than one Pythagorean triangle.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A155177 Area ar/6 (divided by 6) of primitive Pythagorean triangles such that perimeters 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.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A381005 Ordered short legs of the Pythagorean triangles defined by a = 2^(4n) + 2^(2n+1), b = 2^(4n) - 2^(4n-2) - 2^(2n) - 1, c = 2^(4n) + 2^(4n-2) + 2^(2n) + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Formula

A081925 Ordered even short legs of primitive Pythagorean triangles.

Views

Author

Keywords

Links

Crossrefs

Extensions

A120211 x values giving the smallest integer solutions of y^2 = x*(a^N - x)*( b^N + x) (elliptic curve, Weierstrass equation) with a and b legs in primitive Pythagorean triangles and N = 2. Sequence ordered in increasing values of leg a. Relevant y values in A120210.

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Maple

A155178 Numbers p 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.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A024355 Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence gives values of AUB, sorted.

Views

Author

Keywords

Comments

Links

Crossrefs

A024357 Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence gives values of AUBUC, sorted.

Views

Author

Keywords

Links

Crossrefs

A155176 Perimeter s/6 (divided by 6) of primitive Pythagorean triangles such that perimeters are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2pq, s=a+b+c, s-+1 are primes.

A155177 Area ar/6 (divided by 6) of primitive Pythagorean triangles such that perimeters are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2pq, ar=a*b/2; s=a+b+c, s-+1 are primes.

A120211 x values giving the smallest integer solutions of y^2 = x(a^N - x)( b^N + x) (elliptic curve, Weierstrass equation) with a and b legs in primitive Pythagorean triangles and N = 2. Sequence ordered in increasing values of leg a. Relevant y values in A120210.

A155178 Numbers p 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=2pq, ar=ab/2; s=a+b+c, s-+1 are primes, pr=ab*c, pr-+1 are primes.