A024364 -id:A024364 - cp's OEIS frontend

A120089 Square perimeters of primitive Pythagorean triangles.

144, 900, 3136, 8100, 17424, 23716, 33124, 43264, 54756, 57600, 93636, 115600, 139876, 144400, 166464, 174724, 207936, 213444, 244036, 298116, 304704, 357604, 414736, 422500, 476100, 490000, 541696, 571536, 640000, 722500, 746496, 756900

Offset: 1

Lekraj Beedassy, Jun 07 2006

nonn

Square entries of A024364.

Charlie Neder, Table of n, a(n) for n = 1..1000
P. Yiu, "Primitive Pythagorean triangles with square perimeters" in 'Recreational Mathematics' Chap. 6.2 pp. 50/360

Cf. A120090.

isA024364 := proc(an) local r::integer,s::integer ; for r from floor((an/4)^(1/2)) to floor((an/2)^(1/2)) do for s from r-1 to 1 by -2 do if 2*r*(r+s) = an and gcd(r,s) < 2 then RETURN(true) ; fi ; if 2*r*(r+s) < an then break ; fi ; od ; od : RETURN(false) ; end : isA120089 := proc(an) RETURN( issqr(an) and isA024364(an)) ; end: for n from 2 to 1200 do if isA120089(n^2) then printf("%d,",n^2) ; fi ; od ; # R. J. Mathar, Jun 08 2006

A078926[n_] := Sum[Boole[n < d^2 < 2n && CoprimeQ[d, n/d]], {d, Divisors[n/2^IntegerExponent[n, 2]]}];
Reap[For[k = 2, k <= 10^6, k += 2, If[A078926[k/2] > 0 && IntegerQ@Sqrt@k, Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Oct 25 2023 *)

a(n) = (2*u*v)^2, where u=sqrt(j/2) and v=sqrt(j+k) {for coprime pairs (j,k),j>k with odd k such that pairs (u,v),u

a(n) = A024364(k) = A000290(j) for some k and j. - R. J. Mathar, Jun 08 2006

Corrected and extended by R. J. Mathar, Jun 08 2006

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

A322181 Triangle T(n, k) read by rows, n > 0 and 0 < k <= 3^(n-1): T(n, k) = A321768(n, k) + A321769(n, k) + A321770(n, k).

Original entry on oeis.org

12, 30, 70, 40, 56, 176, 126, 208, 408, 198, 154, 234, 84, 90, 330, 260, 546, 1026, 476, 456, 736, 286, 418, 1218, 828, 1178, 2378, 1188, 800, 1160, 390, 340, 900, 570, 644, 1364, 714, 374, 494, 144, 132, 532, 442, 1044, 1924, 874, 918, 1518, 608, 1116, 3196

Offset: 1

Rémy Sigrist, Nov 30 2018

This sequence gives the perimeters 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 A024364.

			The first rows are:
   12
   30, 70, 40
   56, 176, 126, 208, 408, 198, 154, 234, 84
T(1,1) corresponds to the perimeter of the triangle with sides 3, 4, 5; hence T(1, 1) = 3 + 4 + 5 = 12.

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

Cf. A001542, A002939, A024364, A033586, A321768, A321769, A321770.

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] + t[3, 1])

Empirically:
- T(n, 1) = A002939(n+1),
- T(n, (3^(n-1) + 1)/2) = A001542(n+1),
- T(n, 3^(n-1)) = A033586(n).

A376608 Sides x < y < z of Pythagorean triangles ordered first by increasing perimeter x+y+z, then by shorter leg x.

Original entry on oeis.org

3, 4, 5, 6, 8, 10, 5, 12, 13, 9, 12, 15, 8, 15, 17, 12, 16, 20, 7, 24, 25, 10, 24, 26, 15, 20, 25, 20, 21, 29, 18, 24, 30, 16, 30, 34, 12, 35, 37, 21, 28, 35, 9, 40, 41, 15, 36, 39, 24, 32, 40, 27, 36, 45, 14, 48, 50, 20, 48, 52, 24, 45, 51, 30, 40, 50, 28, 45, 53, 11, 60, 61, 33, 44, 55

Offset: 1

Hugo Pfoertner, Sep 29 2024

			   Triangle
   |  Perimeter
   |       x   y   z
   1  12 [ 3,  4,  5]
   2  24 [ 6,  8, 10]
   3  30 [ 5, 12, 13]
   4  36 [ 9, 12, 15]
   5  40 [ 8, 15, 17]
   6  48 [12, 16, 20]
   7  56 [ 7, 24, 25]
   8  60 [10, 24, 26]
   9  60 [15, 20, 25]
  10  70 [20, 21, 29]

Cf. A010814, A024364, A103605.

A374597 uses this order of sides.

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

A308378 Numbers k such that phi(2k+1) = phi(2k+2).

Original entry on oeis.org

0, 1, 7, 127, 247, 487, 1312, 1627, 1852, 2593, 5857, 6682, 9157, 11467, 12772, 23107, 24607, 24667, 28822, 32767, 82087, 92317, 99157, 107887, 143497, 153697, 159637, 194122, 198742, 207637, 245767, 284407, 294703, 343492, 420127

Offset: 1

Torlach Rush, May 24 2019

nonn

For n > 0, 2*a(n) + 1 is a term of A020884. This is because 2*a(n) + 1 is odd and every odd number is the difference of the squares of two consecutive numbers and hence are coprime.
For n > 0, (2*a(n) + 1) * (2*a(n) + 2) is a term of A024364. This is because (2*a(n) + 1) * (2*a(n) + 2) = 2*((a(n) + 1)^2 + (a(n) + 1) * a(n)) and gcd((a(n) + 1), a(n)) = 1.
For n > 0, a(n) is congruent to 1 or 4 mod 6.
2*a(n) + 1 is congruent to 1 or 3 mod 6 and is a term of A047241.
2*a(n) + 2 is congruent to 2 or 4 mod 6 and is a term of A047235.

			0 is a term because phi(1) = phi(2) = 1.
1 is a term because phi(3) = phi(4) = 2.
7 is a term because phi(15) = phi(16) = 8.

Amiram Eldar, Table of n, a(n) for n = 1..5416 (calculated using the b-file at A001274)

Cf. A000010, A004767, A020884, A024364, A047235, A047241, A299535.
Subset of A047234.
Subset of A001274.

Select[Range[0, 9999], EulerPhi[2# + 1] == EulerPhi[2# + 2] &] (* Alonso del Arte, Jul 05 2019 *)
Select[(#-1)/2&/@SequencePosition[EulerPhi[Range[900000]],{x_,x_}][[All,1]],IntegerQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 24 2019 *)

lista(nn) = for(n=0, nn, if(eulerphi(2*n+1) == eulerphi(2*n+2), print1(n, ", ")));
lista(430000)

a(n) = (A299535(n) - 2) / 2.

A328499 The number of primitive Pythagorean triangles with perimeter less than n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 1

Mo Li, Oct 17 2019

D. N. Lehmer has proved that the asymptotic density of a(n) is a(n)/n = log(2)/Pi^2 = 0.07023049... See A118858.

			For n=90, the triples are
   {3,  4,  5},  3 +  4 +  5 = 12 < 90
   {5, 12, 13},  5 + 12 + 13 = 30 < 90
   {7, 24, 25},  7 + 24 + 25 = 56 < 90
   {8, 15, 17},  8 + 15 + 17 = 40 < 90
   {9, 40, 41},  9 + 40 + 41 = 90
  {12, 35, 37}, 12 + 35 + 37 = 84 < 90
  {20, 21, 29}, 20 + 21 + 29 = 70 < 90
so a(90)=7.

Ron Knott, Pythagorean Triples and Online Calculators
D. N. Lehmer, Asymptotic evaluation of certain totient sums, Amer. J. Math. 22, 293-335, 1900.

Cf. A008846, A024364, A118858, A249750.

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, ...

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

A223857 Ordered products of the perimeter and the sides of primitive Pythagorean triangles.

Original entry on oeis.org

720, 23400, 81600, 235200, 852600, 1305360, 1328400, 5314320, 8414280, 9434880, 16893240, 18498480, 33918720, 43995600, 45561600, 46652760, 57757440, 106226640, 108617760, 154736400, 155263680, 184041000, 235227600, 361712400, 417740400, 451760400, 471711240

Offset: 1

Mihir Mathur, Apr 02 2013

Considering the set of primitive Pythagorean triangles with sides (A, B, C), the sequence gives the values (A+B+C)*(A*B*C), in increasing order.

It is a challenge to find a pair of primitive Pythagorean triangles such that product of perimeter and the sides is equal.

			a(1) = (3+4+5)*(3*4*5) = 720.
a(2) = (5+12+13)*(5*12*13) = 23400.

Wikipedia, Pythagorean Triples

Cf. A063011, A024364.

Corrected and extended by Giovanni Resta, Apr 03 2013

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cp's OEIS Frontend

A120089 Square perimeters of primitive Pythagorean triangles.

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A155185 Primes in A155175.

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A322181 Triangle T(n, k) read by rows, n > 0 and 0 < k <= 3^(n-1): T(n, k) = A321768(n, k) + A321769(n, k) + A321770(n, k).

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A376608 Sides x < y < z of Pythagorean triangles ordered first by increasing perimeter x+y+z, then by shorter leg x.

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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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Mathematica

A308378 Numbers k such that phi(2k+1) = phi(2k+2).

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Mathematica

PARI

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A328499 The number of primitive Pythagorean triangles with perimeter less than n.

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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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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.