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URL: https://oeis.org/A046080

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A046080
a(n) is the number of integer-sided right triangles with hypotenuse n.
57
0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 4, 0, 0, 1, 0, 1, 0, 0, 1, 1, 2, 0, 0, 1, 0, 1, 0, 1, 0, 0, 4, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0
OFFSET
1,25
COMMENTS
Or number of ways n^2 can be written as the sum of two positive squares: a(5) = 1: 3^2 + 4^2 = 5^2; a(25) = 2: 7^2 + 24^2 = 15^2 + 20^2 = 25^2. - Alois P. Heinz, Aug 01 2019
Or number of ways n^2 is arithmetically equidistant from two other squares: a(5) = 1: 7^2 - 5^2 = 5^2 - 1^2 (= A256418(1) = 24); a(25) = 2: 31^2 - 25^2 = 25^2 - 17^2 (= A256418(6) = 336) and 35^2 - 25^2 = 25^2 - 5^2 (= A256418(9) = 600). Equivalent for n rather than n^2 gives A025441. - Charles L. Hohn, Dec 31 2025
REFERENCES
A. H. Beiler, Recreations in the Theory of Numbers, New York: Dover, pp. 116-117, 1966.
LINKS
A. Tripathi, On Pythagorean triples containing a fixed integer, Fib. Q., 46/47 (2008/2009), 331-340. See Theorem 7.
Eric Weisstein's World of Mathematics, Pythagorean Triple
FORMULA
Let n = 2^e_2 * product_i p_i^f_i * product_j q_j^g_j where p_i == 1 mod 4, q_j == 3 mod 4; then a(n) = (1/2)*(product_i (2*f_i + 1) - 1). - Beiler, corrected
8*a(n) + 4 = A046109(n) for n > 0. - Ralf Stephan, Mar 14 2004
a(n) = 0 for n in A004144. - Lekraj Beedassy, May 14 2004
a(A084645(k)) = 1. - Ruediger Jehn, Jan 14 2022
a(A084646(k)) = 2. - Ruediger Jehn, Jan 14 2022
a(A084647(k)) = 3. - Jean-Christophe Hervé, Dec 01 2013
a(A084648(k)) = 4. - Jean-Christophe Hervé, Dec 01 2013
a(A084649(k)) = 5. - Jean-Christophe Hervé, Dec 01 2013
a(n) = A063725(n^2) / 2. - Michael Somos, Mar 29 2015
a(n) = Sum_{k=1..n} Sum_{i=1..k} [i^2 + k^2 = n^2], where [ ] is the Iverson bracket. - Wesley Ivan Hurt, Dec 10 2021
a(A002144(k)^n) = n. - Ruediger Jehn, Jan 14 2022
From Ridouane Oudra, Jan 16 2026: (Start)
a(n) = Sum_{d|n} A024362(d).
a(n) = Sum_{d|n} A323239(d)*A025441(n/d).
a(n) = Sum_{d|n} A035263(d)*A157228(n/d).
a(n) = A328712(n) + A024362(n).
a(n) = A063014(n) - 1.
a(n) = (A256452(n) - 1)/2.
a(n) = A092147(A000265(n))/2. (End)
MAPLE
f:= proc(n) local F, t;
F:= select(t -> t[1] mod 4 = 1, ifactors(n)[2]);
1/2*(mul(2*t[2]+1, t=F)-1)
end proc:
map(f, [$1..100]); # Robert Israel, Jul 18 2016
MATHEMATICA
a[1] = 0; a[n_] := With[{fi = Select[ FactorInteger[n], Mod[#[[1]], 4] == 1 & ][[All, 2]]}, (Times @@ (2*fi+1)-1)/2]; Table[a[n], {n, 1, 99}] (* Jean-François Alcover, Feb 06 2012, after first formula *)
PROG
(PARI) a(n)={my(m=0, k=n, n2=n*n, k2, l2);
while(1, k=k-1; k2=k*k; l2=n2-k2; if(l2>k2, break); if(issquare(l2), m++)); return(m)} \\ brute force, Stanislav Sykora, Mar 18 2015
(PARI) {a(n) = if( n<1, 0, sum(k=1, sqrtint(n^2 \ 2), issquare(n^2 - k^2)))}; /* Michael Somos, Mar 29 2015 */
(PARI) a(n) = {my(f = factor(n/(2^valuation(n, 2)))); (prod(k=1, #f~, if ((f[k, 1] % 4) == 1, 2*f[k, 2] + 1, 1)) - 1)/2; } \\ Michel Marcus, Mar 08 2016
(Python)
from math import prod
from sympy import factorint
def A046080(n): return prod((e<<1)+1 for p, e in factorint(n).items() if p&3==1)>>1 # Chai Wah Wu, Sep 06 2022
CROSSREFS
Partial sums: A224921.
A088111 gives records; A088959 gives where records occur.
First differs from A083025 at n=65.
Sequence in context: A088950 A267113 A083025 * A170967 A035227 A355320
KEYWORD
nonn
STATUS
approved