A046081 -id:A046081 - cp's OEIS frontend

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

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.

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);

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).

A088978 Number of Pythagorean triangles having the n-th prime prime(n) as one of their sides.

Original entry on oeis.org

0, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1

Offset: 1

Lekraj Beedassy, Oct 31 2003

nonn

Primitive Pythagorean triples are given parametrically by (M^2 - N^2)^2 + (2MN)^2 = (M^2 + N^2)^2. Odd primes are uniquely representable (ignoring signs) as M^2 - N^2, but only primes of the form 4k + 1 are uniquely representable as M^2 + N^2. Since 2MN is composite for MN > 1, an odd prime can be a side of one or two Pythagorean triangles. Thus, except for a(1) = 0, a(n) is 2 for prime(n) of the form 4k + 1, and 1 otherwise. - Chris Boyd, Jan 25 2016

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

Cf. A046081.

[0] cat [(4-NthPrime(n) mod 4+1)/2: n in [2..100]]; // Vincenzo Librandi, Jan 26 2016

0, seq((4-ithprime(i) mod 4 + 1)/2, i=2..1000); # Robert Israel, Jan 25 2016

Table[(4 - Mod[Prime@ n, 4] + 1)/2, {n, 105}] /. Rational -> 0 (* _Michael De Vlieger, Jan 26 2016 *)

a088978(n) = my(p=prime(n)); if(p==2,0,if((p-1)%4==0,2,1))
for(i=1,105,print1(a088978(i),", ")) \\ Chris Boyd, Jan 25 2016

Corrected and extended by Ray Chandler, Nov 01 2003

A269929 Records for the numbers of Pythagorean triples to which an integer belongs.

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 8, 10, 14, 23, 32, 38, 41, 53, 68, 71, 95, 113, 122, 158, 159, 203, 206, 221, 284, 287, 338, 341, 365, 368, 473, 476, 479, 608, 611, 662, 665, 743, 854, 1016, 1097, 1421, 1430, 1826, 1835, 1988, 2231, 2369, 2564, 2636, 3050, 3293, 3314, 4265, 4274, 5480, 5966, 6695, 7109, 7667, 7910, 9125, 9134

Offset: 1

Michel Marcus, Mar 08 2016

nonn

A. Tripathi, On Pythagorean triples containing a fixed integer, Fib. Q., 46/47 (2008/2009), 331-340. See Theorem 11.
Index entries related to Pythagorean Triples.

Cf. A046081, A269928.

nbpt(n) = {oddn = n/(2^valuation(n, 2)); f = factor(oddn); for (k=1, #f~, if ((f[k,1] % 4) != 1, f[k,2] = 0);); n1 = factorback(f); if (n % 2, (numdiv(n^2)+numdiv(n1^2))/2 -1, (numdiv((n/2)^2)+numdiv(n1^2))/2 -1);}
lista(nn) = {last = -1; for (n=1, nn, if ((new = nbpt(n)) > last, print1(new, ", "); last = new;););}

a(n) = A046081(A269928(n)).

More terms from Jinyuan Wang, Mar 15 2019

A328949 Number of non-primitive Pythagorean triples with n as a leg or the hypotenuse.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 0, 2, 0, 1, 3, 2, 0, 2, 0, 3, 2, 1, 0, 5, 2, 2, 2, 2, 0, 5, 0, 3, 2, 2, 3, 5, 0, 1, 3, 6, 0, 4, 0, 2, 6, 1, 0, 8, 1, 4, 3, 3, 0, 3, 3, 5, 2, 2, 0, 10, 0, 1, 5, 4, 4, 4, 0, 3, 2, 5, 0, 10, 0, 2, 7, 2, 2, 5, 0, 9, 3, 2, 0, 9, 4, 1, 3, 5, 0, 8, 3, 2, 2, 1, 3, 11, 0, 2, 5, 7

Offset: 1

Rui Lin, Nov 01 2019

nonn

Pythagorean triples including primitive ones and non-primitive ones. For a certain n, it may be a leg or the hypotenuse in either a primitive Pythagorean triple, or a non-primitive Pythagorean triple, or both.

This sequence is the count of n as a leg or the hypotenuse in non-primitive Pythagorean triples.

			For n=10, 10 is a leg in (10,24,26) and the hypotenuse in (6,8,10), so a(10)=A328708(10)+A328712(10)=1+1=2. And 10 is not a leg or the hypotenuse in any primitive Pythagorean triple, a(10)=A046081(10)-A024363(10)=2-0=2.

A. Beiler, Recreations in the Theory of Numbers. New York: Dover Publications, pp. 116-117, 1966.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 5000 terms from Metin Sariyar)

Cf. A328708, A328712, A046081, A024363.

a[n_] := Count[{x, y} /. Solve[(x^2 + y^2 == n^2 || x^2 - y^2 == n^2) && x > y > 0, {x, y}, Integers], p_ /; GCD @@ p > 1]; Array[a, 100] (* Giovanni Resta, Nov 01 2019 *)

a(n) = A328708(n) + A328712(n).

a(n) = A046081(n) - A024363(n).

A006593 Least number which is side of n Pythagorean triples.

Original entry on oeis.org

3, 5, 16, 12, 15, 125, 24, 40, 75, 48, 80, 72, 84, 60, 65536, 192, 144, 524288, 384, 640, 9375, 168, 120, 300, 1536, 520, 576, 3072, 975, 2147483648, 336, 240, 1171875, 1500, 1040, 137438953472, 504, 360, 600, 672, 420, 4608, 50000, 780, 3456, 196608, 9216

Offset: 1

N. J. A. Sloane

nonn

First occurrence of n in A046081. - Lekraj Beedassy, Oct 31 2003

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 114.
Jean Meeus, Letter to N. J. A. Sloane, Dec 26 1974
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ron Knott, Pythagorean Triples and Online Calculators
J. Meeus & N. J. A. Sloane, Correspondence, 1974-1975
E. S. Rowland, Pythagorean Triples
Eric Weisstein's World of Mathematics, Pythagorean Triple
Index entries related to Pythagorean Triples.

Cf. A046081.

Corrected and extended by Ray Chandler, Mar 05 2004

A056138 Number of ways in which n can be the shorter leg (shortest side) of an integer-sided right triangle.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 2, 1, 2, 1, 3, 3, 1, 1, 4, 2, 1, 3, 3, 1, 3, 1, 3, 4, 1, 3, 5, 1, 1, 4, 5, 1, 3, 1, 3, 5, 1, 1, 7, 2, 2, 4, 3, 1, 3, 3, 5, 4, 1, 1, 9, 1, 1, 5, 4, 4, 4, 1, 3, 4, 3, 1

Offset: 1

Henry Bottomley, Jun 15 2000

nonn

Cf. A009004, A055525, A055527.

a(n)=my(b);sum(c=n+2,n^2\2+1,issquare(c^2-n^2,&b) && nCharles R Greathouse IV, Jul 07 2013

a(n) = A046079(n) - A056137(n) = A046081(n) - A046080(n) - A056137(n).

A269928 Integers n that belong to more Pythagorean triples than preceding integers.

Original entry on oeis.org

1, 3, 5, 12, 15, 24, 40, 48, 60, 120, 240, 360, 420, 720, 840, 1560, 1680, 2520, 3360, 5040, 8400, 9240, 10920, 15120, 18480, 21840, 27720, 32760, 36960, 43680, 55440, 65520, 109200, 110880, 120120, 166320, 196560, 221760, 240240, 360360, 480480, 720720, 1113840

Offset: 1

Michel Marcus, Mar 08 2016

nonn

Called optimal P-numbers in Tripathi link.

A. Tripathi, On Pythagorean triples containing a fixed integer, Fib. Q., 46/47 (2008/2009), 331-340. See Theorem 11.
Index entries related to Pythagorean Triples.

Cf. A046081, A269929.

nbpt(n) = {oddn = n/(2^valuation(n, 2)); f = factor(oddn); for (k=1, #f~, if ((f[k,1] % 4) != 1, f[k,2] = 0);); n1 = factorback(f); if (n % 2, (numdiv(n^2)+numdiv(n1^2))/2 -1, (numdiv((n/2)^2)+numdiv(n1^2))/2 -1);}
lista(nn) = {last = -1; for (n=1, nn, if ((new = nbpt(n)) > last, print1(n, ", "); last = new;););}

A374845 The numbers p or 2p with p prime and p = 3 mod 4, in ascending order.

Original entry on oeis.org

3, 6, 7, 11, 14, 19, 22, 23, 31, 38, 43, 46, 47, 59, 62, 67, 71, 79, 83, 86, 94, 103, 107, 118, 127, 131, 134, 139, 142, 151, 158, 163, 166, 167, 179, 191, 199, 206, 211, 214, 223, 227, 239, 251, 254, 262, 263, 271, 278, 283, 302, 307, 311, 326, 331, 334, 347, 358, 359, 367, 379, 382, 383, 398

Offset: 1

Manfred Boergens, Jul 22 2024

nonn

Numbers appearing exactly once in a Pythagorean triple and as the smallest number in this triple.
Subsequence of A292762.
Inserting 4 as second term gives A374846.

A. Tripathi, On Pythagorean triples containing a fixed integer, Fib. Q., 46/47 (2008/2009), 331-340. See Theorem 8.

Cf. A374846, A292762, A046081.

t={}; Do[If[(PrimeQ[n]&&Mod[n, 4] == 3)||(PrimeQ[n/2]&&Mod[n/2, 4] == 3), t=Join[t,{n}]], {n, 470}]; t
(* Positions of the ones in  A046081, omitting position = 4;  based on program by Jean-François Alcover *)
a[1] = 0; a[n_] := Module[{f}, f = Select[FactorInteger[n], Mod[#[[1]], 4] == 1 &][[All, 2]]; (DivisorSigma[0, If[OddQ[n], n, n/2]^2] - 1)/2 + (Times @@ (2*f + 1) - 1)/2]; arr = Array[a, nmax]; fl = Flatten[Position[arr, 1]]; Delete[fl, 2]

cp's OEIS Frontend

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

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

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A088978 Number of Pythagorean triangles having the n-th prime prime(n) as one of their sides.

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A269929 Records for the numbers of Pythagorean triples to which an integer belongs.

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A328949 Number of non-primitive Pythagorean triples with n as a leg or the hypotenuse.

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A006593 Least number which is side of n Pythagorean triples.

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A056138 Number of ways in which n can be the shorter leg (shortest side) of an integer-sided right triangle.

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PARI

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A269928 Integers n that belong to more Pythagorean triples than preceding integers.

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