A055523 -id:A055523 - cp's OEIS frontend

A055527 Shortest other leg of a Pythagorean triangle with n as length of a leg.

4, 3, 12, 8, 24, 6, 12, 24, 60, 5, 84, 48, 8, 12, 144, 24, 180, 15, 20, 120, 264, 7, 60, 168, 36, 21, 420, 16, 480, 24, 44, 288, 12, 15, 684, 360, 52, 9, 840, 40, 924, 33, 24, 528, 1104, 14, 168, 120, 68, 39, 1404, 72, 48, 33, 76, 840, 1740, 11, 1860, 960, 16, 48, 72

Offset: 3

Henry Bottomley, May 22 2000

nonn

From Alex Ratushnyak, Mar 30 2014: (Start)
Least positive k such that n^2 + k^2 is a square.
For odd n, a(n) <= 4*triangular((n-1)/2), because n^2 + (4 * triangular((n-1)/2))^2 = ((n^2+1)/2) ^ 2, which is a perfect square since n is odd.
For n = 4*k+2, a(n) <= 8*triangular(k), because (4k+2)^2 + (4*k*(k+1))^2 = (4*k^2 + 4*k + 2)^2. (End)

T. D. Noe, Table of n, a(n) for n = 3..1000

Cf. A000290, A000217, A009112, A046079, A046080, A046081, A054435, A054436, A055522, A055523, A055524, A055525, A055526.

See A082183 for a similar sequence involving triangular numbers.

Table[k = 1; While[! IntegerQ[Sqrt[n^2 + k^2]], k++]; k, {n, 3, 100}] (* T. D. Noe, Apr 02 2014 *)

a(n) = sqrt(A055526(n)^2-n^2) = 2*A054436/n.

A084921 a(n) = lcm(p-1, p+1) where p is the n-th prime.

Original entry on oeis.org

3, 4, 12, 24, 60, 84, 144, 180, 264, 420, 480, 684, 840, 924, 1104, 1404, 1740, 1860, 2244, 2520, 2664, 3120, 3444, 3960, 4704, 5100, 5304, 5724, 5940, 6384, 8064, 8580, 9384, 9660, 11100, 11400, 12324, 13284, 13944, 14964, 16020, 16380

Offset: 1

Reinhard Zumkeller, Jun 11 2003

nonn

This sequence consists of terms of sequences A055523 and A055527 for prime n > 2. - Toni Lassila (tlassila(AT)cc.hut.fi), Feb 02 2004

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Christian Krause, LODA program for A084921
Index entries for sequences related to lcm's

Cf. A000040, A006093, A008864, A009286, A024700, A055523, A055527, A084920, A084922.

a084921 n = lcm (p - 1) (p + 1)  where p = a000040 n
-- Reinhard Zumkeller, Jun 01 2013

[3] cat [(p^2-1)/2: p in PrimesInInterval(3,300)]; // G. C. Greubel, May 03 2024

LCM[#-1,#+1]&/@Prime[Range[50]] (* Harvey P. Dale, Oct 09 2018 *)

a(n)=if(n<2,3,(prime(n)^2-1)/2) \\ Charles R Greathouse IV, May 15 2013

[3]+[(n^2-1)/2 for n in prime_range(3,301)] # G. C. Greubel, May 03 2024

a(n) = A084920(n)/2 for n > 1.
a(n) = 3*A084922(n) for n > 2.
a(n) = A009286(A000040(n)). - Enrique Pérez Herrero, May 17 2012
a(n) ~ 0.5 n^2 log^2 n. - Charles R Greathouse IV, May 15 2013
Product_{n>=1} (1 + 1/a(n)) = 2. - Amiram Eldar, Jan 23 2021
a(n) = (A000040(n)^2 - 1) / 2 for n > 1. - Christian Krause, Mar 27 2021
a(n) = (3/2)*A024700(n-2), for n > 1. - G. C. Greubel, May 03 2024

A055524 Longest other side of a Pythagorean triangle with n as length of one of the three sides (in fact n is a leg and a(n) the hypotenuse).

Original entry on oeis.org

5, 5, 13, 10, 25, 17, 41, 26, 61, 37, 85, 50, 113, 65, 145, 82, 181, 101, 221, 122, 265, 145, 313, 170, 365, 197, 421, 226, 481, 257, 545, 290, 613, 325, 685, 362, 761, 401, 841, 442, 925, 485, 1013, 530, 1105, 577, 1201, 626, 1301, 677, 1405, 730, 1513, 785

Offset: 3

Henry Bottomley, May 22 2000

Paolo Xausa, Table of n, a(n) for n = 3..10000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A009112, A046079, A046080, A046081, A054435, A054436, A055522, A055523, A055525, A055526, A055527.

A055524[n_] := (3*n^2-(-1)^n*(n^2-2)+6)/8; Array[A055524, 100, 3] (* or *)
LinearRecurrence[{0, 3, 0, -3, 0, 1}, {5, 5, 13, 10, 25, 17}, 100] (* Paolo Xausa, Feb 29 2024 *)

Vec(-x^3*(2*x^5+x^4-5*x^3-2*x^2+5*x+5)/((x-1)^3*(x+1)^3) + O(x^100)) \\ Colin Barker, Sep 15 2014

a(n) = sqrt(n^2+A055523(n)^2). a(2k) = k^2+1, a(2k+1) = k^2+(k+1)^2.
a(n) = 3*a(n-2)-3*a(n-4)+a(n-6). G.f.: -x^3*(2*x^5+x^4-5*x^3-2*x^2+5*x+5) / ((x-1)^3*(x+1)^3). - Colin Barker, Sep 15 2014
a(n) = (3*n^2+6-(n^2-2)*(-1)^n)/8. - Luce ETIENNE, Jul 11 2015

A055525 Shortest other side of a Pythagorean triangle having n as length of one of the three sides.

Original entry on oeis.org

4, 3, 3, 8, 24, 6, 12, 6, 60, 5, 5, 48, 8, 12, 8, 24, 180, 12, 20, 120, 264, 7, 7, 10, 36, 21, 20, 16, 480, 24, 44, 16, 12, 15, 12, 360, 15, 9, 9, 40, 924, 33, 24, 528, 1104, 14, 168, 14, 24, 20, 28, 72, 33, 33, 76, 40, 1740, 11, 11, 960, 16, 48, 16, 88, 2244, 32, 92, 24

Offset: 3

Henry Bottomley, May 22 2000

nonn

Robert G. Wilson v, Table of n, a(n) for n = 3..10000

Cf. A009112, A046079, A046080, A046081, A054435, A054436, A055522, A055523, A055524, A055526, A055527.

a[n_] := Block[{a, c, k = 1, n2 = n^2}, While[ If[ k > n, !IntegerQ[c = Sqrt[n2 + k^2]], !IntegerQ[c = Sqrt[n2 + k^2]] && !IntegerQ[a = Sqrt[n2 - k^2]]], k++; If[k == n, k++]]; If[ IntegerQ@ c, k, Sqrt[n2 - a^2]]]; (* Robert G. Wilson v, Feb 23 2024 *)

From Robert G. Wilson v, Feb 23 2024: (Start)
sqrt(2*(n-1)) < a(n) < n^2/2.
If n = k*m, then a(n) <= k*a(m). (End)

A055522 Largest area of a Pythagorean triangle with n as length of one of the three sides (in fact as a leg).

Original entry on oeis.org

6, 6, 30, 24, 84, 60, 180, 120, 330, 210, 546, 336, 840, 504, 1224, 720, 1710, 990, 2310, 1320, 3036, 1716, 3900, 2184, 4914, 2730, 6090, 3360, 7440, 4080, 8976, 4896, 10710, 5814, 12654, 6840, 14820, 7980, 17220, 9240, 19866, 10626, 22770, 12144, 25944

Offset: 3

Henry Bottomley, May 22 2000

Cf. A009112, A046079, A046080, A046081, A054435, A054436, A055523, A055524, A055525, A055526, A055527.

seq(piecewise(n mod 2 = 0,n*(n^2-4)/8,n*(n^2-1)/4),n=3..60); # C. Ronaldo

Table[n*(3*(n^2 - 2) - (n^2 + 2)*(-1)^n)/16, {n, 3, 50}] (* Wesley Ivan Hurt, Apr 27 2017 *)

a(n) = n*A055523(n)/2.
a(2k) = k*(k+1)*(k-1), a(2k+1) = k*(k+1)*(2k+1).
O.g.f.: 6*x^3*(x+1+x^2)/((1-x)^4*(1+x)^4). a(2k+1)=A055112(k). a(2k)=A007531(k+1). [R. J. Mathar, Aug 06 2008]
a(n) = n*(3*(n^2-2)-(n^2+2)*(-1)^n)/16. - Luce ETIENNE, Jul 17 2015

A055526 Shortest hypotenuse of a Pythagorean triangle with n as length of a leg.

Original entry on oeis.org

5, 5, 13, 10, 25, 10, 15, 26, 61, 13, 85, 50, 17, 20, 145, 30, 181, 25, 29, 122, 265, 25, 65, 170, 45, 35, 421, 34, 481, 40, 55, 290, 37, 39, 685, 362, 65, 41, 841, 58, 925, 55, 51, 530, 1105, 50, 175, 130, 85, 65, 1405, 90, 73, 65, 95, 842, 1741, 61, 1861, 962, 65

Offset: 3

Henry Bottomley, May 22 2000

nonn

Smallest k>n such that the squarefree part of k+n equals the squarefree part of k-n - Benoit Cloitre, May 26 2002

Paolo Xausa, Table of n, a(n) for n = 3..1000

Cf. A009112, A046079, A046080, A046081, A054435, A054436, A055522, A055523, A055524, A055525, A055527.

core[n_] := core[n] = Times @@ Map[#[[1]]^Mod[#[[2]], 2] &, FactorInteger[n]];
A055526[n_] := Block[{k = n}, While[core[++k+n] != core[k-n]]; k];
Array[A055526, 100, 3] (* Paolo Xausa, Feb 29 2024 *)

for(n=3,105,s=n+1; while(abs(core(s+n)-core(s-n))>0,s++); print1(s,","))

a(n) = sqrt(n^2+A055527(n)^2).

A054435 Smallest area of a Pythagorean triangle with n as length of one of the three sides.

Original entry on oeis.org

6, 6, 6, 24, 84, 24, 54, 24, 330, 30, 30, 336, 54, 96, 60, 216, 1710, 96, 210, 1320, 3036, 84, 84, 120, 486, 294, 210, 216, 7440, 384, 726, 240, 210, 270, 210, 6840, 270, 180, 180, 840, 19866, 726, 486, 12144, 25944, 336, 4116, 336, 540, 480, 630, 1944, 726

Offset: 3

Henry Bottomley, May 22 2000

nonn

Cf. A009112, A046079, A046080, A046081, A054436, A055522, A055523, A055524, A055525, A055526, A055527.

A054436 Smallest area of a Pythagorean triangle with n as length of a leg.

Original entry on oeis.org

6, 6, 30, 24, 84, 24, 54, 120, 330, 30, 546, 336, 60, 96, 1224, 216, 1710, 150, 210, 1320, 3036, 84, 750, 2184, 486, 294, 6090, 240, 7440, 384, 726, 4896, 210, 270, 12654, 6840, 1014, 180, 17220, 840, 19866, 726, 540, 12144, 25944, 336, 4116, 3000, 1734, 1014

Offset: 3

Henry Bottomley, May 22 2000

nonn

Cf. A009112, A046079, A046080, A046081, A054435, A055522, A055523, A055524, A055525, A055526, A055527.

readlib(issqr): for a from 3 to 80 do for b from 1 by 1 while not issqr(a^2+b^2) do od: printf("%d, ",a*b/2) od: # C. Ronaldo

a[n_] := For[k = 1, True, k++, If[IntegerQ[Sqrt[n^2+k^2]], Return[n k/2]]];
a /@ Range[3, 100] (* Jean-François Alcover, Feb 14 2020 *)

a(n) = n*A055527(n)/2.

A216244 a(n) = (prime(n)^2 - 1)/2 for n >= 2.

Original entry on oeis.org

4, 12, 24, 60, 84, 144, 180, 264, 420, 480, 684, 840, 924, 1104, 1404, 1740, 1860, 2244, 2520, 2664, 3120, 3444, 3960, 4704, 5100, 5304, 5724, 5940, 6384, 8064, 8580, 9384, 9660, 11100, 11400, 12324, 13284, 13944, 14964, 16020, 16380, 18240, 18624, 19404, 19800

Offset: 2

Richard R. Forberg, May 28 2013

Subsequence of A055523 restricted to the case of the other (shorter) leg of the triangle equal to a prime.
There is only one value of a(n) for each prime(n). (This is not necessarily true if the shorter leg is not a prime.)
Note that a(1) is nonexistent since there is no solution with prime = 2.
All terms are divisible by 4.
The values of m (the length of the hypotenuse) always equals a(n) + 1.
a(n) = (prime(n)^2 - 1)/2 for all n > 1.
This follows algebraically given m = a(n) + 1 (or vice versa).
The same two relationships apply when the shorter leg is an odd nonprime, but for only those results corresponding to the longest possible leg of the triangle.

			24^2 + 7^2 = 625 = 25^2 = (24 +1)^2  and a(4) = (prime(4)^2 -1)/2 = (49 - 1)/2 = 24.

Vincenzo Librandi, Table of n, a(n) for n = 2..4000

Subset of A055523.
Equals 4*A061066.
Equals A084921 excluding its first term.

[(NthPrime(n)^2 - 1)/2: n in [2..50]]; // G. C. Greubel, Dec 14 2018

A216244:=n->(ithprime(n)^2-1)/2: seq(A216244(n), n=2..100); # Wesley Ivan Hurt, May 03 2017

Table[(Prime[n]^2 - 1)/2, {n, 2, 100}] (* Vincenzo Librandi, Jun 15 2014 *)

vector(50, n, n++; (prime(n)^2 -1)/2) \\ G. C. Greubel, Dec 14 2018

[(nth_prime(n)^2 -1)/2 for n in (2..50)] # G. C. Greubel, Dec 14 2018

a(n) = (prime(n)^2 - 1)/2 for n >= 2.
a(n) = 4*A061066(n) = A084920(n)/2.
a(n) = A084921(n) for n > 1.
a(n) = (prime(n)-1)*(prime(n)+1)/2 = lcm(prime(n)+1, prime(n)-1) for n > 1 because one of prime(n)+1 or prime(n)-1 is even and the other is divisible by 4. Say prime(n)-1 is divisible by 4; then (prime(n)+1)/2 and (prime(n)-1)/4 must be coprime. - Frank M Jackson, Dec 11 2018
Product_{n>=2} (1 + 1/a(n)) = 3/2. - Amiram Eldar, Jun 03 2022

New name (taken from Formula entry) from Jon E. Schoenfield, Jul 11 2021

A056137 Number of ways in which n can be the longer leg (middle side) of an integer-sided right triangle.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 3, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 2, 0, 0, 0, 2, 0, 1, 0, 1, 2, 0, 0, 3, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 4, 0, 0, 2, 1, 0, 0, 0, 1, 0, 1, 0

Offset: 1

Henry Bottomley, Jun 15 2000

nonn

Cf. A009023, A055523.

a(n) = A046079(n) - A056138(n) = A046081(n) - A046080(n) - A056138(n).

cp's OEIS Frontend

A055527 Shortest other leg of a Pythagorean triangle with n as length of a leg.

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A084921 a(n) = lcm(p-1, p+1) where p is the n-th prime.

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A055524 Longest other side of a Pythagorean triangle with n as length of one of the three sides (in fact n is a leg and a(n) the hypotenuse).

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A055525 Shortest other side of a Pythagorean triangle having n as length of one of the three sides.

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A055522 Largest area of a Pythagorean triangle with n as length of one of the three sides (in fact as a leg).

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A055526 Shortest hypotenuse of a Pythagorean triangle with n as length of a leg.

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A054435 Smallest area of a Pythagorean triangle with n as length of one of the three sides.

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A054436 Smallest area of a Pythagorean triangle with n as length of a leg.

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A216244 a(n) = (prime(n)^2 - 1)/2 for n >= 2.

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