A046079 -id:A046079 - cp's OEIS frontend

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

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)

A349082 The number of two-term Egyptian fractions of rational numbers, x/y, 0 < x/y < 1, ordered as below. The sequence is the number of (p,q) pairs such that x/y = 1/p + 1/q where p and q are integers with p < q.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 1, 0, 4, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 3, 2, 2, 1, 1, 1, 0, 2, 2, 1, 1, 1, 1, 0, 0, 4, 1, 2, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 7, 4, 2, 1, 2, 1, 2, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 4, 1, 3, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 4, 4, 1, 3, 1, 1, 0, 2, 1, 1, 0, 0, 0, 0, 4, 3, 2, 2, 1, 2, 0, 1, 1, 1, 0, 1, 0, 0, 0

Offset: 1

Jud McCranie, Nov 07 2021

The sequence are the terms in a triangle, where the rows correspond to the denominator of the rational number (starting with row 2, column 1) and the columns correspond to the numerators:
                    x=1  2  3  4  5       rationals x/y:
  Row 1 (y=2):  1                   1/2
  Row 2 (y=3):  1, 1                1/3, 2/3
  Row 3 (y=4):  2, 1, 1             1/4, 2/4, 3/4
  Row 4 (y=5):  1, 1, 1, 0          1/5, 2/5, 3/5, 4/5
  Row 5 (y=6):  4, 1, 1, 1, 1       1/6, 2/6, 3/6, 4/6, 5/6
Alternatively, order the rational numbers, x/y, 0 < x/y < 1, in this order: 1/2, 1/3, 2/3, 1/4, 2/4, 3/4, 1/5, 2/5, ... For example, in this ordering, the sixth rational number is 3/4. The numerators of the n-th rational number are A002260(n) and the denominators are A003057(n).
A018892 is a subsequence (for x/y = 1/n).

			The fourth rational number is 1/4, 1/4 = 1/5 + 1/20 = 1/6 + 1/12, so a(4)=2.

Jud McCranie, Table of n, a(n) for n = 1..990

Cf. A002260, A003057.

Columns: A018892 (x=1), A046079 (x=2).

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.

A309507 Number of ways the n-th triangular number T(n) = A000217(n) can be written as the difference of two positive triangular numbers.

Original entry on oeis.org

0, 1, 1, 1, 3, 3, 1, 2, 5, 3, 3, 3, 3, 7, 3, 1, 5, 5, 3, 7, 7, 3, 3, 5, 5, 7, 7, 3, 7, 7, 1, 3, 7, 7, 11, 5, 3, 7, 7, 3, 7, 7, 3, 11, 11, 3, 3, 5, 8, 11, 7, 3, 7, 15, 7, 7, 7, 3, 7, 7, 3, 11, 5, 3, 15, 7, 3, 7, 15, 7, 5, 5, 3, 11, 11, 7, 15, 7, 3, 9, 9, 3, 7

Offset: 1

Alois P. Heinz, Aug 05 2019

nonn

Equivalently, a(n) is the number of triples [n,k,m] with k>0 satisfying the Diophantine equation n*(n+1) + k*(k+1) - m*(m+1) = 0. Any such triple satisfies a triangle inequality, n+k > m. The n for which there is a triple [n,n,m] are listed in A053141. - Bradley Klee, Mar 01 2020; edited by N. J. A. Sloane, Mar 31 2020

			a(5) = 3: T(5) = T(6)-T(3) = T(8)-T(6) = T(15)-T(14).
a(7) = 1: T(7) = T(28)-T(27).
a(8) = 2: T(8) = T(13)-T(10) = T(36)-T(35).
a(9) = 5: T(9) = T(10)-T(4) = T(11)-T(6) = T(16)-T(13) = T(23)-T(21) = T(45)-T(44).
a(49) = 8: T(49) = T(52)-T(17) = T(61)-T(36) = T(94)-T(80) = T(127)-T(117) = T(178)-T(171) = T(247)-T(242) = T(613)-T(611) = T(1225)-T(1224).
The triples with n <= 16 are:
2, 2, 3
3, 5, 6
4, 9, 10
5, 3, 6
5, 6, 8
5, 14, 15
6, 5, 8
6, 9, 11
6, 20, 21
7, 27, 28
8, 10, 13
8, 35, 36
9, 4, 10
9, 6, 11
9, 13, 16
9, 21, 23
9, 44, 45
10, 8, 13
10, 26, 28
10, 54, 55
11, 14, 18
11, 20, 23
11, 65, 66
12, 17, 21
12, 24, 27
12, 77, 78
13, 9, 16
13, 44, 46
13, 90, 91
14, 5, 15
14, 11, 18
14, 14, 20
14, 18, 23
14, 33, 36
14, 51, 53
14, 104, 105
15, 21, 26
15, 38, 41
15, 119, 120
16, 135, 136. - _N. J. A. Sloane_, Mar 31 2020

Alois P. Heinz, Table of n, a(n) for n = 1..20000
J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, Three Cousins of Recaman's Sequence, arXiv:2004:14000 [math.NT], April 2020.
M. A. Nyblom, On the representation of the integers as a difference of nonconsecutive triangular numbers, Fibonacci Quarterly 39:3 (2001), pp. 256-263.

Cf. A000217, A001108, A046079 (the same for squares), A068194, A100821 (the same for primes for n>1), A309332.
See also A053141. The monotonic triples [n,k,m] with n <= k <= m are counted in A333529.
Cf. A092517, A063440.

with(numtheory): seq(tau(n*(n+1))-tau(n*(n+1)/2)-1, n=1..80); # Ridouane Oudra, Dec 08 2023

TriTriples[TNn_] := Sort[Select[{TNn, (TNn + TNn^2 - # - #^2)/(2 #),
      (TNn + TNn^2 - # + #^2)/(2 #)} & /@
    Complement[Divisors[TNn (TNn + 1)], {TNn}],
   And[And @@ (IntegerQ /@ #), And @@ (# > 0 & /@ #)] &]]
Length[TriTriples[#]] & /@ Range[100]
(* Bradley Klee, Mar 01 2020 *)

a(n) = 1 <=> n in { A068194 } \ { 1 }.
a(n) is even <=> n in { A001108 } \ { 0 }.
a(n) = number of odd divisors of n*(n+1) (or, equally, of T(n)) that are greater than 1. - N. J. A. Sloane, Apr 03 2020
a(n) = A092517(n) - A063440(n) - 1. - Ridouane Oudra, Dec 08 2023

A379830 a(n) is the number of Pythagorean triples (u, v, w) for which w - u = n where u < v < w.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 2, 2, 1, 0, 1, 0, 1, 0, 2, 0, 4, 0, 1, 0, 1, 0, 2, 3, 1, 2, 1, 0, 1, 0, 5, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 1, 2, 1, 0, 2, 4, 7, 0, 1, 0, 4, 0, 2, 0, 1, 0, 1, 0, 1, 2, 5, 0, 1, 0, 1, 0, 1, 0, 8, 0, 1, 3, 1, 0, 1, 0, 2, 6, 1, 0, 1, 0, 1, 0

Offset: 0

Felix Huber, Jan 07 2025

nonn

The difference between the hypotenuse and the short leg of a primitive Pythagorean triple (p^2 - q^2, 2*p*q, p^2 + q^2) (where p > q are coprimes and not both odd) is d = max(2*q^2, (p - q)^2). For every of these primitive Pythagorean triples whose d divides n, there is a Pythagorean triple with w - u = n. Therefore d <= n and it follows that 1 <= q <= sqrt(n/2) and q + 1 <= p <= q + sqrt(n), which means that there is a finite number of Pythagorean triples with w - u = n.

			The a(18) = 4 Pythagorean triples are (27, 36, 45), (16, 30, 34), (40, 42, 58), (7, 24, 25) because 45 - 27 = 34 - 16 = 58 - 40 = 25 - 7 = 18.
See also linked Maple program "Pythagorean triples for which w - u = n".

Felix Huber, Table of n, a(n) for n = 0..10000
Felix Huber, Pythagorean triples for which w - u = n
Eric Weisstein's World of Mathematics, Pythagorean Triple

Cf. A024361, A024362, A024363, A046079, A046080, A046081, A096033, A224921.

A379830:=proc(n)
    local a,p,q;
    a:=0;
    for q to isqrt(floor(n/2)) do
        for p from q+1 to q+isqrt(n) do
            if igcd(p,q)=1 and (is(p,even) or is(q,even)) and n mod max((p-q)^2,2*q^2)=0 then
                a:=a+1
            fi
        od
    od;
    return a
end proc;
seq(A379830(n),n=0..87);

A221837 Number of integer Heron triangles of height n such that the angles adjacent to the base are not right.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 4, 4, 1, 1, 16, 1, 1, 16, 9, 1, 4, 1, 16, 16, 1, 1, 49, 4, 1, 9, 16, 1, 16, 1, 16, 16, 1, 16, 49, 1, 1, 16, 49, 1, 16, 1, 16, 49, 1, 1, 100, 4, 4, 16, 16, 1, 9, 16, 49, 16, 1, 1, 169, 1, 1, 49, 25, 16, 16, 1, 16, 16, 16, 1, 144, 1, 1, 49

Offset: 1

Eric M. Schmidt, Jan 27 2013

nonn

The number of integer Heron triangles of height n with a right angle at the base is given by A046079.

a(n) is the number of distinct triangles with height n that can be formed with two right triangles with integer sides, either by joining them or by cutting off the smaller one from the larger one. In both cases, the two right triangles must have a leg of length n. To form a triangle with height n, there are binomial(A046079(n) + 1, 2) ways to join them and binomial(A046079(n), 2) ways to cut off the smaller one from the larger one. That's a total of (A046079(n)+1, 2) + (A046079(n), 2)= (A046079(n))^2. - Felix Huber, Aug 20 2023

			The triangle for n = 3 has side lengths (5, 5, 8) and area 12.

Eric M. Schmidt, Table of n, a(n) for n = 1..10000
Sourav Sen Gupta, Nirupam Kar, Subhamoy Maitra, Santanu Sarkar, and Pantelimon Stanica, Counting Heron triangles with Constraints, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 13, Paper A3, 2013.
Eric Weisstein's World of Mathematics, Heronian Triangle.

Cf. A046079, A221838.

def A221837(n) : return (number_of_divisors(n^2 if n%2==1 else (n/2)^2) - 1)^2 // 4

a(n) = A046079(n)^2.

A221838 Number of integer Heron triangles of height n.

Original entry on oeis.org

0, 0, 2, 2, 2, 2, 2, 6, 6, 2, 2, 20, 2, 2, 20, 12, 2, 6, 2, 20, 20, 2, 2, 56, 6, 2, 12, 20, 2, 20, 2, 20, 20, 2, 20, 56, 2, 2, 20, 56, 2, 20, 2, 20, 56, 2, 2, 110, 6, 6, 20, 20, 2, 12, 20, 56, 20, 2, 2, 182, 2, 2, 56, 30, 20, 20, 2, 20, 20, 20, 2, 156, 2, 2

Offset: 1

Eric M. Schmidt, Jan 27 2013

nonn

			For n = 3, the two triangles have side lengths (3, 4, 5) and (5, 5, 8), with areas 6 and 12 respectively.

Eric M. Schmidt, Table of n, a(n) for n = 1..10000
Sourav Sen Gupta, Nirupam Kar, Subhamoy Maitra, Santanu Sarkar, and Pantelimon Stanica, Counting Heron triangles with Constraints, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 13, Paper A3, 2013.
Eric Weisstein's World of Mathematics, Heronian Triangle.

Cf. A046079, A221837.

def A221838(n) : pyth = (number_of_divisors(n^2 if n%2==1 else (n/2)^2) - 1) // 2; return pyth^2 + pyth

a(n) = A221837(n) + A046079(n) = A046079(n)^2 + A046079(n).

cp's OEIS Frontend

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

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A349082 The number of two-term Egyptian fractions of rational numbers, x/y, 0 < x/y < 1, ordered as below. The sequence is the number of (p,q) pairs such that x/y = 1/p + 1/q where p and q are integers with p < q.

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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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A309507 Number of ways the n-th triangular number T(n) = A000217(n) can be written as the difference of two positive triangular numbers.

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A379830 a(n) is the number of Pythagorean triples (u, v, w) for which w - u = n where u < v < w.

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A221837 Number of integer Heron triangles of height n such that the angles adjacent to the base are not right.

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