

A024361


Number of primitive Pythagorean triangles with leg n.


12



0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 2, 1, 1, 0, 1, 2, 2, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, 2, 2, 1, 0, 1, 2, 2, 0, 1, 2, 1, 0, 2, 2, 1, 0, 2, 2, 2, 0, 1, 4, 1, 0, 2, 1, 2, 0, 1, 2, 2, 0, 1, 2, 1, 0, 2, 2, 2, 0, 1, 2, 1, 0, 1, 4, 2, 0, 2, 2, 1, 0, 2, 2, 2, 0, 2, 2, 1, 0, 2, 2, 1, 0, 1, 2, 4
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OFFSET

1,12


COMMENTS

Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence gives number of times A or B takes value n.
For n > 1, a(n) = 0 for n == 2 (mod 4) (n in A016825).
From Jianing Song, Apr 23 2019: (Start)
Note that all the primitive Pythagorean triangles are given by A = min{2*u*v, u^2  v^2}, B = max{2*u*v, u^2  v^2}, C = u^2 + v^2, where u, v are coprime positive integers, u > v and u  v is odd. As a result:
(a) if n is odd, then a(n) is the number of representations of n to the form n = u^2  v^2, where u, v are coprime positive integers (note that this guarantees that u  v is odd) and u > v. Let s = u + v, t = u  v, then n = s*t, where s and t are unitary divisors of n and s > t, so the number of representations is A034444(n)/2 if n > 1 and 0 if n = 1;
(b) if n is divisible by 4, then a(n) is the number of representations of n to the form n = 2*u*v, where u, v are coprime positive integers (note that this also guarantees that u  v is odd because n/2 is even) and u > v. So u and v must be unitary divisors of n/2, so the number of representations is A034444(n/2)/2. Since n is divisible by 4, A034444(n/2) = A034444(n) so a(n) = A034444(n)/2.
(c) if n == 2 (mod 4), then n/2 is odd, so n = 2*u*v implies that u and v are both odd, which is not acceptable, so a(n) = 0.
a(n) = 0 if n = 1 or n == 2 (mod 4), otherwise a(n) is a power of 2.
The earliest occurrence of 2^k is 2*A002110(k+1) for k > 0. (End)


REFERENCES

Tripathi, Amitabha, On Pythagorean triples containing a fixed integer. Fibonacci Quart. 46/47 (2008/09), no. 4, 331340.


LINKS

Antti Karttunen, Table of n, a(n) for n = 1..20000
Ron Knott, Pythagorean Triples and Online Calculators
J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, Three Cousins of Recaman's Sequence, arXiv:2004:14000, April 2020
Eric Weisstein's World of Mathematics, Pythagorean Triple


FORMULA

a(n) = A034444(n)/2 = 2^(A001221(n)1) if n != 2 (mod 4) and n > 1, a(n) = 0 otherwise.  Jianing Song, Apr 23 2019
a(n) = A024359(n) + A024360(n).  Ray Chandler, Feb 03 2020


EXAMPLE

a(12) = 2 because 12 appears twice, in (A,B,C) = (5,12,13) and (12,35,37).


MATHEMATICA

Table[If[n == 1  Mod[n, 4] == 2, 0, 2^(Length[FactorInteger[n]]  1)], {n, 100}]


PROG

(PARI) A024361(n) = if(1==n(2==(n%4)), 0, 2^(omega(n)1)); \\ (after the Mathematica program)  Antti Karttunen, Nov 10 2018


CROSSREFS

Cf. A024359, A024360, A024362, A024363, A046079, A020883, A020884, A034444, A046079.
Sequence in context: A096419 A283616 A130182 * A305614 A190676 A329308
Adjacent sequences: A024358 A024359 A024360 * A024362 A024363 A024364


KEYWORD

nonn


AUTHOR

David W. Wilson


EXTENSIONS

Incorrect comment removed by Ant King, Jan 28 2011
More terms from Antti Karttunen, Nov 10 2018


STATUS

approved



