A009003 -id:A009003 - cp's OEIS frontend

A224921 Number of Pythagorean triples (a, b, c) with a^2 + b^2 = c^2 and 0 < a < b < c < n.

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, 8, 9, 9, 9, 10, 11, 11, 11, 11, 12, 13, 13, 14, 14, 15, 16, 17, 17, 17, 17, 18, 18, 18, 18, 18, 20, 21, 22, 23, 23, 24, 24, 24, 25, 25, 26, 27, 27, 27, 27, 31, 31, 31, 32, 32, 33, 33, 33

Offset: 1

Reiner Moewald, Apr 19 2013

nonn

a(n+1) > a(n) iff n is in A009003. - Benoit Cloitre, Dec 08 2021

Reiner Moewald and Robert Israel, Table of n, a(n) for n = 1..10000 (n = 1..500 from Reiner Moewald)

Cf. A156685. Essentially partial sums of A046080.

Cf. A009003.

a046080:= 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:
ListTools:-PartialSums(map(a046080, [$0..100])); # Robert Israel, Jul 18 2016

b[0] = b[1] = 0; b[n_] := With[{fi = Select[FactorInteger[n], Mod[#[[1]], 4] == 1&][[All, 2]]}, (Times @@ (2*fi + 1) - 1)/2];
Table[b[n], {n, 0, 100}] // Accumulate (* Jean-François Alcover, Feb 27 2019 *)

a(n)=sum(a=1,n-3,sum(b=a+1,sqrtint((n-1)^2-a^2), issquare(a^2+b^2))) \\ Charles R Greathouse IV, Apr 29 2013

A242118 Number of unit squares that intersect the circumference of a circle of radius n centered at (0,0).

Original entry on oeis.org

0, 4, 12, 20, 28, 28, 44, 52, 60, 68, 68, 84, 92, 92, 108, 108, 124, 124, 140, 148, 148, 164, 172, 180, 188, 180, 196, 212, 220, 220, 228, 244, 252, 260, 260, 268, 284, 284, 300, 300, 308, 316, 332, 340, 348, 348, 364, 372, 380, 388, 380

Offset: 0

Kival Ngaokrajang, May 05 2014

nonn

For the points that form the Pythagorean triple (for example see illustration n = 5, on the first quadrant at coordinate (4,3) and (3,4)), the transit of circumference occurs exactly at the corners, therefore there are no additional intersecting squares on the upper or lower rows (diagonally NE & SW directions) of such points.

If the center of the circle is instead chosen at the middle of a square grid centered at (1/2,0), the sequence will be 2*A004767(n-1).

Kival Ngaokrajang, Illustration of initial terms

Cf. A009003, A004767.

a = lambda n: sum(4 for x in range(n) for y in range(n)
                    if x**2 + y**2 < n**2 and (x+1)**2 + (y+1)**2 > n**2)

from sympy import factorint
def a(n):
    r = 1
    for p, e in factorint(n).items():
        if p%4 == 1: r *= 2*e + 1
    return 8*n - 4*r if n > 0 else 0

a(n) = 4*Sum{k=1..n} ceiling(sqrt(n^2 - (k-1)^2)) - floor(sqrt(n^2 - k^2)). - Orson R. L. Peters, Jan 30 2017

a(n) = 8*n - A046109(n) for n > 0. - conjectured by Orson R. L. Peters, Jan 30 2017, proved by Andrey Zabolotskiy, Jan 31 2017

Terms corrected by Orson R. L. Peters, Jan 30 2017

A210503 Numbers k that form a primitive Pythagorean triple with k' and sqrt(k^2 + k'^2), where k' is the arithmetic derivative of k.

Original entry on oeis.org

15, 35, 143, 323, 899, 1763, 3599, 4641, 5183, 10403, 11663, 13585, 19043, 22499, 32399, 35581, 36863, 39203, 51983, 57599, 72899, 79523, 97343, 121103, 176399, 186623, 213443, 272483, 324899, 359999, 381923, 412163, 435599, 446641, 622081, 656099, 675683

Offset: 1

Paolo P. Lava, Jan 25 2013

nonn

A037074 is a subsequence of this sequence.
If k is the product of a pair of twin primes we have k=p(p+2), k'=2(p+1) and sqrt(k^2+k'^2)=(p+1)^2+1=p(p+2)+2=k+2. These numbers are relatively prime and therefore they form a primitive Pythagorean triple.
Also in the sequence are the following numbers with four distinct prime factors:
        4641 = 3*7*13*17       [form p(p+4)*q(q+4)],
       13585 = 5*11*13*19      [form p(p+6)*q(q+6)],
       35581 = 7*13*17*23      [form p(p+6)*q(q+6)],
      446641 = 13*17*43*47     [form p(p+4)*q(q+4)],
      622081 = 17*23*37*43     [form p(p+6)*q(q+6)],
      700321 = 19*29*31*41     [form p(p+10)*q(q+10)],
From Ray Chandler, Jan 25 2017: (Start)
    24887581 = 47*53*97*103    [form p(p+6)*q(q+6)],
    43518577 = 59*67*101*109   [form p(p+8)*q(q+8)],
   115539901 = 83*97*113*127   [form p(p+14)*q(q+14)],
   158682817 = 89*101*127*139  [form p(p+12)*q(q+12)],
   305162941 = 103*113*157*167 [form p(p+10)*q(q+10)],
  1093514641 = 103*107*313*317 [form p(p+4)*q(q+4)],
  1415940061 = 167*193*197*223 [form p(p+26)*q(q+26)].
And one term with six distinct prime factors:
   650344079 = 7*11*37*53*59*73. (End)

			m=57599, m'=480, sqrt(57599^2 + 480^2) = 57601.

Ray Chandler, Table of n, a(n) for n = 1..500 (terms 1..100 from Paolo P. Lava)

Cf. A003415, A009003, A009004, A037074.

with(numtheory);
A210503:= proc(q)
local a,n,p;
for n from 1 to q do
  a:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);
  if trunc(sqrt(n^2+a^2))=sqrt(n^2+a^2) and gcd(n,gcd(a,n^2+a^2))=1 then print(n); fi;
od; end:
A210503(100000);

from math import sqrt
from sympy import factorint
from gmpy2 import mpz, is_square, gcd
A210503 = []
for n in range(2, 10**5):
    nd = sum([mpz(n*e/p) for p, e in factorint(n).items()])
    if is_square(nd**2+n**2) and gcd(gcd(n, nd), mpz(sqrt(nd**2+n**2))) == 1:
        A210503.append(n) # Chai Wah Wu, Aug 21 2014

A134422 Square numbers which are sums of 2 distinct nonzero squares.

Original entry on oeis.org

25, 100, 169, 225, 289, 400, 625, 676, 841, 900, 1156, 1225, 1369, 1521, 1600, 1681, 2025, 2500, 2601, 2704, 2809, 3025, 3364, 3600, 3721, 4225, 4624, 4900, 5329, 5476, 5625, 6084, 6400, 6724, 7225, 7569, 7921, 8100, 8281, 9025, 9409, 10000, 10201

Offset: 1

Artur Jasinski, Oct 25 2007

nonn

			25 = 5^2 = 4^2 + 3^2, and so 25 is in the sequence.
100 = 10^2 = 8^2 + 6^2, and so 100 is in the sequence.
169 = 13^2 = 12^2 + 5^2, and so 169 is in the sequence.

Index entries for sequences related to sums of squares

Cf. A000404, A000415, A009003.

c = {}; Do[Do[k = a^2 + b^2; If[IntegerQ[Sqrt[k]], AppendTo[c, k]], {a, 1, b - 1}], {b, 200}]; Union[c] (* Artur Jasinski *)
Select[Range[100]^2, Length[PowersRepresentations[#, 2, 2]] > 1 &] (* Alonso del Arte, Feb 11 2014 *)

select(n->for(k=1,sqrtint(n\2),if(issquare(n-k^2), return(n>k^2)));0, vector(100,i,i^2)) \\ Charles R Greathouse IV, Jul 02 2013

a(n) = A009003(n)^2.

A161882 Smallest k such that n^2 = a_1^2 + ... + a_k^2 and all a_i are positive integers less than n.

Original entry on oeis.org

4, 3, 4, 2, 3, 3, 4, 3, 2, 3, 3, 2, 3, 2, 4, 2, 3, 3, 2, 3, 3, 3, 3, 2, 2, 3, 3, 2, 2, 3, 4, 3, 2, 2, 3, 2, 3, 2, 2, 2, 3, 3, 3, 2, 3, 3, 3, 3, 2, 2, 2, 2, 3, 2, 3, 3, 2, 3, 2, 2, 3, 3, 4, 2, 3, 3, 2, 3, 2, 3, 3, 2, 2, 2, 3, 3, 2, 3, 2, 3, 2, 3, 3, 2, 3, 2, 3, 2, 2, 2, 3, 3, 3, 2, 3, 2, 3, 3, 2, 2, 2, 3, 2, 2, 2

Offset: 2

Dmitry Kamenetsky, Jun 21 2009

nonn

Related to hypotenuse numbers: A161882(A009003(n))=2 for all n.

Jacobi's four-square theorem can be used to show that a(n) <= 4. - Charles R Greathouse IV, Jul 31 2011

			2^2 = 1^2 + 1^2 + 1^2 + 1^2, so a(2)=4.
3^2 = 2^2 + 2^2 + 1^2, so a(3)=3.

Alois P. Heinz, Table of n, a(n) for n = 2..700
Jean-Charles Meyrignac, Computing minimal equal sums of like powers.
Eric Weisstein's World of Mathematics, Diophantine Equation 2nd Powers.

Cf. A161883, A161884, A161885.

f[n_, k_] := Select[PowersRepresentations[n^2, k, 2], AllTrue[#, 0<#Jean-François Alcover, Oct 03 2020 *)

A161882(n)={vecmin(factor(n)[,1]%4)==1 && return(2);  if(n==1<M. F. Hasler, Dec 17 2014

a(n)=2 iff n is in A009003 (hypotenuse numbers), a(n)=4 iff n is in A000079 (powers of 2), otherwise a(n)=3. - M. F. Hasler, Dec 17 2014

More terms from Alois P. Heinz, Dec 04 2014

A267113 Bitwise-OR of the exponents of all 4k+1 primes in the prime factorization of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 03 2016

nonn

			For n = 65 = 5 * 13 = (4+1)^1 * ((3*4)+1)^1, bitwise-or of 1 and 1 is 1, thus a(65) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A170818, A267116, A260728.
Cf. A004144 (indices of zeros), A009003 (of nonzeros).
Differs from both A046080 and A083025 for the first time at n=65, which here a(65) = 1.

a(n) = A267116(A170818(n)).
Other identities. For all n >= 0:
a(n) = a(A170818(n)). [The result depends only on the prime factors of the form 4k+1.]
a(n) <= A083025(n).

A156682 Consider all Pythagorean triangles A^2 + B^2 = C^2 with AA009004(n)).

Original entry on oeis.org

5, 13, 10, 25, 17, 15, 41, 26, 61, 20, 37, 85, 50, 25, 39, 113, 34, 65, 145, 30, 82, 181, 29, 52, 101, 35, 75, 221, 122, 265, 40, 51, 74, 145, 65, 313, 170, 45, 123, 365, 53, 100, 197, 421, 50, 78, 226, 481, 68, 130, 257, 55, 65, 183, 545, 290, 91, 125, 613, 60, 85, 111

Offset: 1

Ant King, Feb 17 2009

The corresponding sequence for primitive triples is A156679. For all triples, the ordered sequence of C values is A020882 (allowing repetitions) and A009003 (excluding repetitions).

			As the first four Pythagorean triples (ordered by increasing A) are (3,4,5), (5,12,13), (6,8,10) and (7,24,25), then a(1)=5, a(2)=13, a(3)=10 and a(4)=25.

Beiler, Albert H.: Recreations In The Theory Of Numbers, Chapter XIV, The Eternal Triangle, Dover Publications Inc., New York, 1964, pp. 104-134.
Sierpinski, W.; Pythagorean Triangles, Dover Publications, Inc., Mineola, New York, 2003.

Ron Knott, Right-angled Triangles and Pythagoras' Theorem

Cf. A009004, A156681, A156679, A020882, A009003.

PythagoreanTriplets[n_]:=Module[{t={{3,4,5}},i=4,j=5},While[i

a(n) = sqrt(A009004(n)^2 + A156681(n)^2).

A230543 Numbers n that form a Pythagorean quadruple with n', n'' and sqrt(n^2 + n'^2 + n''^2), where n' and n'' are the first and the second arithmetic derivative of n.

Original entry on oeis.org

512, 1203, 3456, 6336, 23328, 42768, 157464, 249753, 288684, 400000, 722718, 1062882, 1948617, 2700000, 4950000, 18225000, 33412500, 105413504, 123018750, 225534375, 312500000, 408918816

Offset: 1

Paolo P. Lava, Oct 25 2013

Tested up to n = 4.09*10^8.

			If n = 6336 then n' = 23808, n'' = 103936 and sqrt(n^2 + n'^2 + n''^2) = 106816.

Eric Weisstein's World of Mathematics, Pythagorean Quadruple

Cf. A003415, A009003, A009004, A068346, A210503, A211176.

Cf. A096907-A096909 and A097263-A097266 for Pythagorean Quadruples.

with(numtheory): P:= proc(q) local a1, a2, n, p;
for n from 2 to q do a1:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);
a2:=a1*add(op(2,p)/op(1,p),p=ifactors(a1)[2]);
if type(sqrt(n^2+a1^2+a2^2),integer) then print(n);
fi; od; end: P(10^10);

a(16)-a(18) from Giovanni Resta, Oct 25 2013
a(19) from Ray Chandler, Dec 22 2016
a(20) from Ray Chandler, Dec 31 2016
a(21) from Ray Chandler, Jan 05 2017
a(22) from Ray Chandler, Jan 09 2017

A025302 Numbers that are the sum of 2 distinct nonzero squares in exactly 1 way.

Original entry on oeis.org

5, 10, 13, 17, 20, 25, 26, 29, 34, 37, 40, 41, 45, 50, 52, 53, 58, 61, 68, 73, 74, 80, 82, 89, 90, 97, 100, 101, 104, 106, 109, 113, 116, 117, 122, 136, 137, 146, 148, 149, 153, 157, 160, 164, 169, 173, 178, 180, 181, 193, 194, 197, 200, 202, 208, 212, 218, 225, 226, 229

Offset: 1

David W. Wilson

nonn

From Fermat's two squares theorem, every prime of the form 4k + 1 is a term (A002144). - Bernard Schott, Apr 15 2022

Donovan Johnson, Table of n, a(n) for n = 1..1000.
Art of Problem Solving, Fermat's Two Squares Theorem.
Index entries for sequences related to sums of squares.

Cf. A002144 (subsequence), A009000, A009003, A024507, A025441, A004431.

Cf. Subsequence of A001983; A004435.

a025302 n = a025302_list !! (n-1)
a025302_list = [x | x <- [1..], a025441 x == 1]

nn = 229; t = Table[0, {nn}]; lim = Floor[Sqrt[nn - 1]]; Do[num = i^2 + j^2; If[num <= nn, t[[num]]++], {i, lim}, {j, i - 1}]; Flatten[Position[t, 1]] (* T. D. Noe, Apr 07 2011 *)
a[1] = 5; a[ n_] := a[n] = Module[ {s = a[n - 1], t = True, j}, While[ t, s++; Do[ If[ i^2 + (j = Floor[Sqrt[s - i^2]])^2 == s && i < j, t = False; Break], {i, Sqrt[s/2]}]]; s]; (* Michael Somos, Jan 20 2019 *)

from collections import Counter
from itertools import combinations
def aupto(lim):
  s = filter(lambda x: x <= lim, (i*i for i in range(1, int(lim**.5)+2)))
  s2 = filter(lambda x: x <= lim, (sum(c) for c in combinations(s, 2)))
  s2counts = Counter(s2)
  return sorted(k for k in s2counts if k <= lim and s2counts[k] == 1)
print(aupto(229)) # Michael S. Branicky, May 10 2021

A025441(a(n)) = 1. - Reinhard Zumkeller, Dec 20 2013

A120211 x values giving the smallest integer solutions of y^2 = x(a^N - x)( b^N + x) (elliptic curve, Weierstrass equation) with a and b legs in primitive Pythagorean triangles and N = 2. Sequence ordered in increasing values of leg a. Relevant y values in A120210.

Original entry on oeis.org

4, 6, 12, 24, 15, 40, 60, 40, 70, 84, 72, 56, 126, 144, 180, 168, 198, 180, 220, 264, 126, 286, 312, 364, 360, 390, 420, 480, 510, 49, 544, 300, 612, 616, 646, 684, 720, 760, 288, 798, 840, 924, 726, 966, 700, 1012, 1104, 990, 1150, 1200

Offset: 1

Giorgio Balzarotti, Paolo P. Lava, Jun 10 2006

nonn

			First primitive Pythagorean triad: 3, 4, 5
Weierstrass equation. y^2 = x*( 3^2 - x)*( 4^2 + x)
Smallest integer solution (x, y) = (4,20)
First element in the sequence x = 4

G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 47.

Cf. A009003, A020884, A120210-A120213.

flag :=1;x:=0; # a, b, c primitive Pythagorean triad while flag =1 do x:=x+1; y2:= x*( a^2 - x)*(x+b^2); if ((floor(sqrt(y2)))^2=y2)then print( x);flag :=0;fi; od;

cp's OEIS Frontend

A224921 Number of Pythagorean triples (a, b, c) with a^2 + b^2 = c^2 and 0 < a < b < c < n.

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A242118 Number of unit squares that intersect the circumference of a circle of radius n centered at (0,0).

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A210503 Numbers k that form a primitive Pythagorean triple with k' and sqrt(k^2 + k'^2), where k' is the arithmetic derivative of k.

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A134422 Square numbers which are sums of 2 distinct nonzero squares.

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A161882 Smallest k such that n^2 = a_1^2 + ... + a_k^2 and all a_i are positive integers less than n.

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A267113 Bitwise-OR of the exponents of all 4k+1 primes in the prime factorization of n.

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A156682 Consider all Pythagorean triangles A^2 + B^2 = C^2 with AA009004(n)).

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A230543 Numbers n that form a Pythagorean quadruple with n', n'' and sqrt(n^2 + n'^2 + n''^2), where n' and n'' are the first and the second arithmetic derivative of n.

A120211 x values giving the smallest integer solutions of y^2 = x(a^N - x)( b^N + x) (elliptic curve, Weierstrass equation) with a and b legs in primitive Pythagorean triangles and N = 2. Sequence ordered in increasing values of leg a. Relevant y values in A120210.