A024406 -id:A024406 - cp's OEIS frontend

A057229 a(n) = ab = xy with (a-b) = (x+y) = A020882(n) (a>b, a>0, b>0, x>0, y>0), gcd(a, b) = gcd(x, y) = 1.

6, 30, 60, 84, 210, 210, 180, 630, 330, 504, 924, 1320, 546, 1386, 1560, 2340, 990, 2730, 840, 2574, 4620, 1224, 1716, 3570, 5610, 7140, 4290, 1710, 5016, 7956, 7980, 2730, 7854, 10374, 2310, 11970, 6630, 10920, 12540, 4080, 3036, 11856, 8970

Offset: 0

Naohiro Nomoto, Sep 19 2000

nonn

The quadratics in X, X^2 - S*X -+ P, where S=A020882(n), P=A057229(n) are each factorizable into two factors, all four being distinct: X^2 - S*X - P = (X - a)*(X + b). X^2 - S*X + P = (X - x)*(X - y). - Lekraj Beedassy, Apr 30 2004

Areas of primitive Pythagorean triangles sorted on hypotenuse A020882, then on perimeter A093507. - Lekraj Beedassy, Aug 18 2006

			E.g. a(1)=6=6*1=3*2, (6-1)=(3+2)=5=A020882(1), gcd(6,1)=gcd(3,2)=1

P. Yiu, Factorizable x^2 + px -+ q, Recreational Mathematics, pp. 58/360.

Cf. A020882, A008846, A024406, A024365.

A105520 Sums of area and perimeter of Pythagorean triples, sorted in increasing order, including duplicates.

Original entry on oeis.org

18, 48, 60, 90, 100, 140, 144, 180, 210, 270, 280, 288, 294, 320, 360, 378, 448, 462, 480, 594, 600, 648, 660, 720, 728, 756, 858, 900, 900, 924, 980, 1008, 1008, 1078, 1080, 1120, 1170, 1210, 1260, 1344, 1496, 1530, 1530, 1568, 1584, 1584, 1680, 1700, 1728

Offset: 1

Alexandre Wajnberg, May 02 2005

			a(28) = 900 = (18+80+82) + (18*80/2) for 18*18 + 80*80 = 82*82.
a(29) = 900 = (25+60+65) + (25*60/2) for 25*25 + 60*60 = 65*65.
a(32) = 1008 = (24+70+74) + (24*70/2) for 24*24 + 70*70 = 74*74.
a(33) = 1008 = (36+48+60) + (36*48/2) for 36*36 + 48*48 = 60*60.

Giovanni Resta, Table of n, a(n) for n = 1..10711 (terms < 10^7)
Douglas Butler, Pythagorean triples [including multiples] up to 2100, Oundle School iCT Training Centre.
Ron Knott, Pythagorean triples.

Cf. A103605, A103606, A024364, A024406, A105521.

L = {}; mx = 1728; Do[ Do[ If[ IntegerQ[z = Sqrt[x^2 + y^2]], v = x y/2 + x + y + z; If[v <= mx, AppendTo[L, v], Break[]]], {y, x-1}], {x, 4, 4 + (2 mx^2)^(1/3)}]; Sort@ L (* Giovanni Resta, Mar 16 2020 *)

T. = 0                        ;  S = ''
do C = 1 to 999               ;  H = C*C
   do D = 1 to C              ;  I = D*D
      do E = 1 to D           ;  J = E*E
         if I + J < H   then  iterate E
         if I + J = H   then  do
            K = T.0 + 1       ;  T.0 = K
            P = C + D + E     ;  A = ( D * E ) / 2
            T.K = right( A + P, 6 )
            T.K = T.K '=' A '+' P '(' E '+' D '+' C ')'
         end
         leave E
      end E
   end D
end C
call KWIK 'T.' /* sort by A+P for area A and perimeter P */
Y = 0
do N = 1 to T.0 while length( S ) < 255
   X = word( T.N, 1 )         ;  say T.N
   if X <= Y   then  say 'dupe:' N - 1 N ':' Y X
   S = S || ', ' || X         ;  Y = X
end N
say substr( S, 3 )            /* Frank Ellermann, Mar 02 2020 */

Corrected and extended by Frank Ellermann, Mar 02 2020

A155185 Primes in A155175.

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

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

Hypotenuse C (prime numbers only) 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. p=1,q=2,a=3,b=4,c=5=prime,s=12-+1primes, ...

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 *)

Sequence corrected by Ray Chandler, Feb 11 2020

A322170 Triangle T(n, k) read by rows, n > 0 and 0 < k <= 3^(n-1): T(n, k) = A321768(n, k) * A321769(n, k) / 2.

Original entry on oeis.org

6, 30, 210, 60, 84, 1320, 630, 1560, 7140, 1386, 924, 2340, 210, 180, 4620, 2730, 10920, 45144, 7854, 7980, 23184, 2574, 5016, 63336, 26910, 49476, 242556, 50490, 25200, 57420, 4290, 3570, 34650, 12540, 14490, 79794, 18564, 5610, 10374, 504, 330, 11970, 7956

Offset: 1

Rémy Sigrist, Nov 29 2018

This sequence gives the areas of the primitive Pythagorean triangles corresponding to the primitive Pythagorean triples in the tree described in A321768.

If we order the terms in this sequence and keep duplicates then we obtain A024406.

			The first rows are:
   6
   30, 210, 60
   84, 1320, 630, 1560, 7140, 1386, 924, 2340, 210
T(1,1) corresponds to the area of the triangle with sides 3, 4, 5; hence T(1, 1) = 3 * 4 / 2 = 6.

Rémy Sigrist, Rows n = 1..9, flattened
Index entries related to Pythagorean Triples

Cf. A024406, A029549, A055112, A069072, A321768, A321769.

M = [[1, -2, 2; 2, -1, 2; 2, -2, 3], [1, 2, 2; 2, 1, 2; 2, 2, 3], [-1, 2, 2; -2, 1, 2; -2, 2, 3]];
T(n, k) = my (t=[3; 4; 5], d=digits(3^(n-1)+k-1, 3)); for (i=2, #d, t = M[d[i]+1] * t); return (t[1, 1] * t[2, 1] / 2)

Empirically:
- T(n, 1) = A055112(n),
- T(n, (3^(n-1) + 1)/2) = A029549(n),
- T(n, 3^(n-1)) = A069072(n-1).

A105521 Sums of area and perimeter of primitive Pythagorean triples.

Original entry on oeis.org

18, 60, 100, 140, 270, 280, 294, 462, 648, 728, 756, 1078, 1080, 1210, 1496, 1530, 1584, 1768, 2028, 2090, 2574, 2772, 2860, 2990, 3150, 3588, 3910, 4550, 4624, 4680, 4950, 5434, 5670, 5984, 6498, 6960, 7140, 7548, 8330, 8398, 8432, 8436, 8820, 9568, 10098

Offset: 1

Alexandre Wajnberg, May 02 2005

Douglas Butler, Table of Pythagorean triples up to 2100, Oundle School iCT Training Centre
Ron Knott, Pythagorean triples.

Cf. A103605, A103606, A024364, A024406, A105520

Corrected and extended by Harvey P. Dale, Oct 27 2018

A155186 Primes in A155171.

Original entry on oeis.org

2, 7, 29, 101, 107, 197, 227, 457, 647, 829, 1549, 1627, 2221, 2309, 2347, 2521, 2677, 2801, 3181, 3299, 3529, 3541, 3557, 3739, 3769, 4231, 4549, 4871, 4987, 5651, 5827, 5881, 6037, 6079, 6637, 6827, 7517, 7639, 7937, 9787, 11621, 12041, 12329, 13009

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

Numbers p (prime numbers only) 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.

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173, A155174, A155175, A155176, A155177, A155178, A155180, A088483, A001844, A096891, A066885, A099776, A110494, A081589, A155185, A019389, A062090, A050150

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[p],AppendTo[lst,p]]],{n,8!}];lst

A155187 Prime numbers q of primitive Pythagorean triangles such that perimeters are averages of twin prime pairs, p+1=q(prime), 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

2, 3, 11, 71, 227, 491, 683, 1103, 1187, 2591, 3923, 4271, 4931, 6737, 7193, 7703, 8093, 8753, 8963, 9173, 9377, 10271, 13043, 13451, 13997, 15233, 15443, 15803, 15887, 17957, 18701, 19961, 20681, 21701, 22031, 22073, 24371, 24473, 24683

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

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

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173, A155174, A155175, A155176, A155177, A155178, A155180, A088483, A001844, A096891, A066885, A099776, A110494, A081589, A155185, A019389, A062090, A050150, A155186, A068231, A129517, A054574, A132281.

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[q],AppendTo[lst,q]]],{n,8!}];lst

cp's OEIS Frontend

A057229 a(n) = a*b = x*y with (a-b) = (x+y) = A020882(n) (a>b, a>0, b>0, x>0, y>0), gcd(a, b) = gcd(x, y) = 1.

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A105520 Sums of area and perimeter of Pythagorean triples, sorted in increasing order, including duplicates.

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Rexx

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A155185 Primes in A155175.

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A322170 Triangle T(n, k) read by rows, n > 0 and 0 < k <= 3^(n-1): T(n, k) = A321768(n, k) * A321769(n, k) / 2.

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A105521 Sums of area and perimeter of primitive Pythagorean triples.

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A155186 Primes in A155171.

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A155187 Prime numbers q of primitive Pythagorean triangles such that perimeters are averages of twin prime pairs, p+1=q(prime), 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.

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Mathematica

A057229 a(n) = ab = xy with (a-b) = (x+y) = A020882(n) (a>b, a>0, b>0, x>0, y>0), gcd(a, b) = gcd(x, y) = 1.

A155187 Prime numbers q of primitive Pythagorean triangles such that perimeters are averages of twin prime pairs, p+1=q(prime), a=q^2-p^2, c=q^2+p^2, b=2pq, ar=a*b/2; s=a+b+c, s-+1 are primes.