A004431 -id:A004431 - cp's OEIS frontend

A000404 Numbers that are the sum of 2 nonzero squares.

2, 5, 8, 10, 13, 17, 18, 20, 25, 26, 29, 32, 34, 37, 40, 41, 45, 50, 52, 53, 58, 61, 65, 68, 72, 73, 74, 80, 82, 85, 89, 90, 97, 98, 100, 101, 104, 106, 109, 113, 116, 117, 122, 125, 128, 130, 136, 137, 145, 146, 148, 149, 153, 157, 160, 162, 164, 169, 170, 173, 178

Offset: 1

N. J. A. Sloane and J. H. Conway

From the formula it is easy to see that if k is in this sequence, then so are all odd powers of k. - T. D. Noe, Jan 13 2009
Also numbers whose cubes are the sum of two nonzero squares. - Joe Namnath and Lawrence Sze
A line perpendicular to y=mx has its first integral y-intercept at a^2+b^2. The remaining ones for that slope are multiples of that primitive value. - Larry J Zimmermann, Aug 19 2010
The primes in this sequence are sequence A002313.
Complement of A018825; A025426(a(n)) > 0; A063725(a(n)) > 0. - Reinhard Zumkeller, Aug 16 2011
If the two squares are not equal, then any power is still in the sequence: if k = x^2 + y^2 with x != y, then k^2 = (x^2-y^2)^2 + (2xy)^2 and k^3 = (x(x^2-3y^2))^2 + (y(3x^2-y^2))^2, etc. - Carmine Suriano, Jul 13 2012
There are never more than 3 consecutive terms that differ by 1. Triples of consecutive terms that differ by 1 occur infinitely many times, for example, 2(k^2 + k)^2, (k^2 - 1)^2 + (k^2 + 2 k)^2, and (k^2 + k - 1)^2 + (k^2 + k + 1)^2 for any integer k > 1. - Ivan Neretin, Mar 16 2017 [Corrected by Jerzy R Borysowicz, Apr 14 2017]
Number of terms less than 10^k, k=1,2,3,...: 3, 34, 308, 2690, 23873, 215907, 1984228, ... - Muniru A Asiru, Feb 01 2018
The squares in this sequence are the squares of the so-called hypotenuse numbers A009003. - M. F. Hasler, Jun 20 2025

			25 = 3^2 + 4^2, therefore 25 is a term. Note that also 25^3 = 15625 = 44^2 + 117^2, therefore 15625 is a term.

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.
GCHQ, The GCHQ Puzzle Book, Penguin, 2016. See page 103.
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 75, Theorem 4, with Theorem 2, p. 15.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, p. 219, th. 251, 252.
Ian Stewart, "Game, Set and Math", Chapter 8, 'Close Encounters of the Fermat Kind', Penguin Books, Ed. 1991, pp. 107-124.

T. D. Noe, Table of n, a(n) for n = 1..10000
J. M. De Koninck and V. Ouellet, On the n-th element of a set of positive integers, Annales Univ. Sci. Budapest Sect. Comput. 44 (2015), 153-164. See 2. on p. 162.
Étienne Fouvry, Claude Levesque, and Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.
Joshua Harrington, Lenny Jones, and Alicia Lamarche, Representing integers as the sum of two squares in the ring Z_n, arXiv:1404.0187 [math.NT], 2014.
David Rabahy, Google Sheets
G. Xiao, Two squares.
Reinhard Zumkeller, Illustration for A084888 and A000404.
Index entries for sequences related to sums of squares

A001481 gives another version (allowing for zero squares).
Cf. A004431 (2 distinct squares), A063725 (number of representations), A024509 (numbers with multiplicity), A025284, A018825. Also A050803, A050801, A001105, A033431, A084888, A000578, A000290, A057961, A232499, A007692.
Cf. A003325 (analog for cubes), A003336 (analog for 4th powers).
Cf. A009003 (square roots of the squares in this sequence).
Column k=2 of A336725.
Cf. A355237, A355238.

P:=List([1..10^4],i->i^2);;
A000404 := Set(Flat(List(P, i->List(P, j -> i+j)))); # Muniru A Asiru, Feb 01 2018

import Data.List (findIndices)
a000404 n = a000404_list !! (n-1)
a000404_list = findIndices (> 0) a025426_list
-- Reinhard Zumkeller, Aug 16 2011

lst:=[]; for n in [1..178] do f:=Factorization(n); if IsSquare(n) then for m in [1..#f] do d:=f[m]; if d[1] mod 4 eq 1 then Append(~lst, n); break; end if; end for; else t:=0; for m in [1..#f] do d:=f[m]; if d[1] mod 4 eq 3 and d[2] mod 2 eq 1 then t:=1; break; end if; end for; if t eq 0 then Append(~lst, n); end if; end if; end for; lst; // Arkadiusz Wesolowski, Feb 16 2017

nMax:=178: A:={}: for i to floor(sqrt(nMax)) do for j to floor(sqrt(nMax)) do if i^2+j^2 <= nMax then A := `union`(A, {i^2+j^2}) else  end if end do end do: A; # Emeric Deutsch, Jan 02 2017

nMax=1000; n2=Floor[Sqrt[nMax-1]]; Union[Flatten[Table[a^2+b^2, {a,n2}, {b,a,Floor[Sqrt[nMax-a^2]]}]]]
Select[Range@ 200, Length[PowersRepresentations[#, 2, 2] /. {0, } -> Nothing] > 0 &] (* _Michael De Vlieger, Mar 24 2016 *)
Module[{upto=200},Select[Union[Total/@Tuples[Range[Sqrt[upto]]^2,2]],#<= upto&]] (* Harvey P. Dale, Sep 18 2021 *)

is_A000404(n)= for( i=1,#n=factor(n)~%4, n[1,i]==3 && n[2,i]%2 && return); n && ( vecmin(n[1,])==1 || (n[1,1]==2 && n[2,1]%2)) \\ M. F. Hasler, Feb 07 2009

list(lim)=my(v=List(),x2); lim\=1; for(x=1,sqrtint(lim-1), x2=x^2; for(y=1,sqrtint(lim-x2), listput(v,x2+y^2))); Set(v) \\ Charles R Greathouse IV, Apr 30 2016

from itertools import count, islice
from sympy import factorint
def A000404_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        c = False
        for p in (f:=factorint(n)):
            if (q:= p & 3)==3 and f[p]&1:
                break
            elif q == 1:
                c = True
        else:
            if c or f.get(2,0)&1:
                yield n
A000404_list = list(islice(A000404_gen(),30)) # Chai Wah Wu, Jul 01 2022

Let k = 2^t * p_1^a_1 * p_2^a_2 * ... * p_r^a_r * q_1^b_1 * q_2^b_2 * ... * q_s^b_s with t >= 0, a_i >= 0 for i=1..r, where p_i == 1 (mod 4) for i=1..r and q_j == -1 (mod 4) for j=1..s. Then k is a term iff 1) b_j == 0 (mod 2) for j=1..s and 2) r > 0 or t == 1 (mod 2) (or both).
From Charles R Greathouse IV, Nov 18 2022: (Start)
a(n) ~ k*n*sqrt(log n), where k = 1.3085... = 1/A064533.
There are B(x) = (x/sqrt(log x)) * (K + B2/log x + O(1/log^2 x)) terms of this sequence up to x, where K = A064533 and B2 = A227158. (End)

Edited by Ralf Stephan, Nov 15 2004
Typo in formula corrected by M. F. Hasler, Feb 07 2009
Erroneous Mathematica program fixed by T. D. Noe, Aug 07 2009
PARI code fixed for versions > 2.5 by M. F. Hasler, Jan 01 2013

A009003 Hypotenuse numbers (squares are sums of 2 nonzero squares).

Original entry on oeis.org

5, 10, 13, 15, 17, 20, 25, 26, 29, 30, 34, 35, 37, 39, 40, 41, 45, 50, 51, 52, 53, 55, 58, 60, 61, 65, 68, 70, 73, 74, 75, 78, 80, 82, 85, 87, 89, 90, 91, 95, 97, 100, 101, 102, 104, 105, 106, 109, 110, 111, 113, 115, 116, 117, 119, 120, 122, 123, 125, 130, 135, 136, 137, 140

Offset: 1

David W. Wilson

nonn

Multiples of Pythagorean primes A002144 or of primitive Pythagorean triangles' hypotenuses A008846. - Lekraj Beedassy, Nov 12 2003
This is exactly the sequence of positive integers with at least one prime divisor of the form 4k + 1. Compare A072592. - John W. Layman, Mar 12 2008 and Franklin T. Adams-Watters, Apr 26 2009
Circumradius R of the triangles such that the area, the sides and R are integers. - Michel Lagneau, Mar 03 2012
The 2 squares summing to a(n)^2 cannot be equal because sqrt(2) is not rational. - Jean-Christophe Hervé, Nov 10 2013
Closed under multiplication. The primitive elements are those with exactly one prime divisor of the form 4k + 1 with multiplicity one, which are also those for which there exists a unique integer triangle = A084645. - Jean-Christophe Hervé, Nov 11 2013
a(n) are numbers whose square is the mean of two distinct nonzero squares. This creates 1-to-1 mapping between a Pythagorean triple and a "Mean" triple. If the Pythagorean triple is written, abnormally, as {j, k, h} where j^2 +(j+k)^2 = h^2, and h = a(n), then the corresponding "Mean" triple with the same h is {k, 2j, h} where (k^2 + (k+2j)^2)/2 = h^2. For example for h = 5, the Pythagorean triple is {3, 1, 5} and the Mean triple is {1, 6, 5}. - Richard R. Forberg, Mar 01 2015
Integral side lengths of rhombuses with integral diagonals p and q (therefore also with integral areas A because A = pq/2 is some multiple of 24). No such rhombuses are squares. - Rick L. Shepherd, Apr 09 2017
Conjecture: these are bases n in which exists an n-adic integer x satisfying x^5 = x, and 5 is the smallest k>1 such that x^k =x (so x^2, x^3 and x^4 are not x). Example: the 10-adic integer x = ...499879186432 (A120817) satisfies x^5 = x, and x^2, x^3, and x^4 are not x, so 10 is in this sequence. See also A120817, A210850 and A331548. - Patrick A. Thomas, Mar 01 2020
Didactic comment: When students solve a quadratic equation a*x^2 + b*x + c = 0 (a, b, c: integers) with the solution formula, they often make the mistake of calculating b^2 + 4*a*c instead of b^2 - 4*a*c (especially if a or c is negative). If the root then turns out to be an integer, they feel safe. This sequence lists the absolute values of b for which this error can happen. Reasoning: With p^2 = b^2 - 4*a*c and q^2 = b^2 + 4*a*c it follows by addition immediately that p^2 + q^2 = 2*b^2. If 4*a*c < 0, let p = x + y and q = x - y. If 4*a*c > 0, let p = x - y and q = x + y. In both cases follows that y^2 + x^2 = b^2. So every Pythagorean triple gives an absolute value of b for which this error can occur. Example: From (y, x, b) = (3, 4, 5) follows (q^2, b^2, p^2) = (1, 25, 49) or (p^2, b^2, q^2) = (1, 25, 49) with abs(4*a*c) = 24. - Felix Huber, Jul 22 2023
Conjecture: Numbers m such that the limit: Limit_{s->1} zeta(s)*Sum_{k=1..m} [k|m]*A008683(k)*(i^k)/(k^(s - 1)) exists, which is equivalent to numbers m such that abs(Sum_{k=1..m} [k|m]*A008683(k)*(i^k)) = 0. - Mats Granvik, Jul 06 2024

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 98-104.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
R. Chapman, Pythagorean triples and sums of squares
Steven R. Finch, Landau-Ramanujan Constant [Broken link]
Steven R. Finch, Landau-Ramanujan Constant [From the Wayback machine]
Ron Knott, Pythagorean Triples and Online Calculators
J. Pahikkala, On contraharmonic mean and Pythagorean triples, Elemente der Mathematik, 65:2 (2010), 62-67.
Patrick A. Thomas, Solutions of x^5=x up to base 100
Index entries for sequences related to sums of squares

Cf. A000404 (sums of 2 squares), A004431 (sums of 2 distinct squares), A009000 (hypotenuse numbers with repetition), A072592, A004613, A187811.
Complement of A004144. Primes in this sequence give A002144. Same as A146984 (integer contraharmonic means) as sets - see Pahikkala 2010, Theorem 5.
Cf. A083025, A084645 (primitive elements), A084646, A084647, A084648, A084649, A006339.

import Data.List (findIndices)
a009003 n = a009003_list !! (n-1)
a009003_list = map (+ 1) $ findIndices (> 0) a005089_list
-- Reinhard Zumkeller, Jan 07 2013

isA009003 := proc(n)
    local p;
    for p in numtheory[factorset](n) do
        if modp(p,4) = 1 then
            return true;
        end if;
    end do:
    false;
end proc:
for n from 1 to 200 do
    if isA009003(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Nov 17 2014

f[n_] := Module[{k = 1}, While[(n - k^2)^(1/2) != IntegerPart[(n - k^2)^(1/2)], k++; If[2 * k^2 >= n, k = 0; Break[]]]; k]; A009003 = {}; Do[If[f[n^2] > 0, AppendTo[A009003, n]], {n, 3, 100}]; A009003 (* Vladimir Joseph Stephan Orlovsky, Jun 15 2009 *)
Select[Range[200], Length[PowersRepresentations[#^2, 2, 2]] > 1 &] (* Alonso del Arte, Feb 11 2014 *)

is_A009003(n)=setsearch(Set(factor(n)[,1]%4),1)  \\ M. F. Hasler, May 27 2012

list(lim)=my(v=List(),u=vectorsmall(lim\=1)); forprimestep(p=5,lim,4, forstep(n=p,lim,p, u[n]=1)); for(i=5,lim, if(u[i], listput(v,i))); u=0; Vec(v) \\ Charles R Greathouse IV, Jan 13 2022

from itertools import count, islice
from sympy import primefactors
def A009003_gen(): # generator of terms
    return filter(lambda n:any(map(lambda p: p % 4 == 1,primefactors(n))),count(1))
A009003_list = list(islice(A009003_gen(),20)) # Chai Wah Wu, Jun 22 2022

A005089(a(n)) > 0. - Reinhard Zumkeller, Jan 07 2013
a(n) ~ n. - Charles R Greathouse IV, Jan 13 2022
a(n) = sqrt(n-th square in A000404), where A000404 lists the sums of two nonzero squares. - M. F. Hasler, Jun 20 2025

Definition edited by Jean-Christophe Hervé, Nov 10 2013

A008846 Hypotenuses of primitive Pythagorean triangles.

Original entry on oeis.org

5, 13, 17, 25, 29, 37, 41, 53, 61, 65, 73, 85, 89, 97, 101, 109, 113, 125, 137, 145, 149, 157, 169, 173, 181, 185, 193, 197, 205, 221, 229, 233, 241, 257, 265, 269, 277, 281, 289, 293, 305, 313, 317, 325, 337, 349, 353, 365, 373, 377, 389, 397, 401, 409, 421, 425, 433

Offset: 1

N. J. A. Sloane, Ralph Peterson (RALPHP(AT)LIBRARY.nrl.navy.mil)

Numbers of the form x^2 + y^2 where x is even, y is odd and gcd(x, y)=1. Essentially the same as A004613.
Numbers n for which there is no solution to 4/n = 2/x + 1/y for integers y > x > 0. Related to A073101. - T. D. Noe, Sep 30 2002
Discovered by Frénicle (on Pythagorean triangles): Méthode pour trouver ..., page 14 on 44. First text of Divers ouvrages ... Par Messieurs de l'Académie Royale des Sciences, in-folio, 6+518+1 pp., Paris, 1693. Also A020882 with only one of doubled terms (first: 65). - Paul Curtz, Sep 03 2008
All divisors of terms are of the form 4*k+1 (products of members of A002144). - Zak Seidov, Apr 13 2011
A024362(a(n)) > 0. - Reinhard Zumkeller, Dec 02 2012
Closed under multiplication. Primitive elements are in A002144. - Jean-Christophe Hervé, Nov 10 2013
Not only the square of these numbers is equal to the sum of two nonzero squares, but the numbers themselves also are; this sequence is then a subsequence of A004431. - Jean-Christophe Hervé, Nov 10 2013
Conjecture: numbers p for which sqrt(-1) exists in the p-adic numbering system. For example the 5-adic number ...2431212, when squared, gives ...4444444, which is -1, and 5 is in the sequence. - Thierry Banel, Aug 19 2022
The above conjecture was proven true by George Bergman. 3 known facts: (1) prime factors of a(n) are equal to 1 mod 4, (2) modulo such primes, sqrt(-1) exists, (3) if sqrt(m) exists mod r, r being odd, this extends to sqrt(m) in the r-adic ring. - Thierry Banel, Jul 04 2025

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, pp. 10, 107.

Zak Seidov, Table of n, a(n) for n = 1..87881 (with a(n) up to 10^6).
Ron Knott, Pythagorean Triples and Online Calculators

Subsequence of A004431 and of A000404 and of A339952; primitive elements: A002144.
Cf. A004613, A020882, A073101.
Cf. A137409 (complement), disjoint union of A024409 and A120960.

a008846 n = a008846_list !! (n-1)
a008846_list = filter f [1..] where
   f n = all ((== 1) . (`mod` 4)) $ filter ((== 0) . (n `mod`)) [1..n]
-- Reinhard Zumkeller, Apr 27 2011

for x from 1 by 2 to 50 do for y from 2 by 2 to 50 do if gcd(x,y) = 1 then print(x^2+y^2); fi; od; od; [ then sort ].

Union[ Map[ Plus@@(#1^2)&, Select[ Flatten[ Array[ {2*#1, 2*#2-1}&, {10, 10} ], 1 ], GCD@@#1 == 1& ] ] ] (* Olivier Gérard, Aug 15 1997 *)
lst = {}; Do[ If[ GCD[m, n] == 1, a = 2 m*n; b = m^2 - n^2; c = m^2 + n^2; AppendTo[lst, c]], {m, 100}, {n, If[ OddQ@m, 2, 1], m - 1, 2}]; Take[ Union@ lst, 57] (* Robert G. Wilson v, May 02 2009 *)
Union[Sqrt[#[[1]]^2+#[[2]]^2]&/@Union[Sort/@({Times@@#,(Last[#]^2-First[#]^2)/2}&/@ (Select[Subsets[Range[1,33,2],{2}],GCD@@#==1&]))]] (* Harvey P. Dale, Aug 26 2012 *)

is(n)=Set(factor(n)[,1]%4)==[1] \\ Charles R Greathouse IV, Nov 06 2015

# for an array from the beginning
from math import gcd, isqrt
hypothenuses_upto = 433
A008846 = set()
for x in range(2, isqrt(hypothenuses_upto)+1):
    for y in range(min(x-1, (yy:=isqrt(hypothenuses_upto-x**2))-(yy%2 == x%2)) , 0, -2):
        if gcd(x,y) == 1: A008846.add(x**2 + y**2)
print(A008846:=sorted(A008846)) # Karl-Heinz Hofmann, Sep 30 2024

# for single k
from sympy import factorint
def A008846_isok(k): return not any([(pf-1) % 4 for pf in factorint(k)]) # Karl-Heinz Hofmann, Oct 01 2024

x^2 + y^2 where x is even, y is odd and gcd(x, y)=1. Essentially the same as A004613.

More terms from T. D. Noe, Sep 30 2002

A024670 Numbers that are sums of 2 distinct positive cubes.

Original entry on oeis.org

9, 28, 35, 65, 72, 91, 126, 133, 152, 189, 217, 224, 243, 280, 341, 344, 351, 370, 407, 468, 513, 520, 539, 559, 576, 637, 728, 730, 737, 756, 793, 854, 855, 945, 1001, 1008, 1027, 1064, 1072, 1125, 1216, 1241, 1332, 1339, 1343, 1358, 1395, 1456, 1512, 1547, 1674

Offset: 1

Clark Kimberling

nonn

This sequence contains no primes since x^3+y^3=(x^2-x*y+y^2)*(x+y). - M. F. Hasler, Apr 12 2008
There are no terms == 3, 4, 5 or 6 mod 9. - Robert Israel, Oct 07 2014
a(n) mod 2: {1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,1,1,0, ...} - Daniel Forgues, Sep 27 2018

			9 is in the sequence since 2^3 + 1^3 = 9.
35 is in the sequence since 3^3 + 2^3 = 35.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..902 from M. F. Hasler)
Index to sequences related to sums of cubes

See also: Sums of 2 positive cubes (not necessarily distinct): A003325. Sums of 3 distinct positive cubes: A024975. Sums of distinct positive cubes: A003997. Sums of 2 distinct nonnegative cubes: A114090. Sums of 2 nonnegative cubes: A004999. Sums of 2 distinct positive squares: A004431. Cubes: A000578.
Cf. A373971 (characteristic function).
Indices of nonzero terms in A025468.

N:= 10000: # to get all terms <= N
S:= select(`<=`,{seq(seq(i^3 + j^3, j = 1 .. i-1), i = 2 .. floor(N^(1/3)))},N);
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(S,list));
# Robert Israel, Oct 07 2014

lst={};Do[Do[x=a^3;Do[y=b^3;If[x+y==n,AppendTo[lst,n]],{b,Floor[(n-x)^(1/3)],a+1,-1}],{a,Floor[n^(1/3)],1,-1}],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 22 2009 *)
Select[Range@ 1700, Total@ Boole@ Map[And[! MemberQ[#, 0], UnsameQ @@ #] &, PowersRepresentations[#, 2, 3]] > 0 &] (* Michael De Vlieger, May 13 2017 *)

isA024670(n)=for( i=ceil(sqrtn( n\2+1,3)),sqrtn(n-.5,3), isA000578(n-i^3) & return(1)) /* One could also use "for( i=2,sqrtn( n\2-1,3),...)" but this is much slower since there are less cubes in [n/2,n] than in [1,n/2]. Replacing the -1 here by +.5 would yield A003325, allowing for a(n)=x^3+x^3. Replacing -1 by 0 may miss some a(n) of this form due to rounding errors. - M. F. Hasler, Apr 12 2008 */

from itertools import count, takewhile
def aupto(limit):
    cbs = list(takewhile(lambda x: x <= limit, (i**3 for i in count(1))))
    sms = set(c+d for i, c in enumerate(cbs) for d in cbs[i+1:])
    return sorted(s for s in sms if s <= limit)
print(aupto(1674)) # Michael S. Branicky, Sep 28 2021

Name edited by Zak Seidov, May 31 2011

A009000 Ordered hypotenuse numbers (squares are sums of 2 distinct nonzero squares).

Original entry on oeis.org

5, 10, 13, 15, 17, 20, 25, 25, 26, 29, 30, 34, 35, 37, 39, 40, 41, 45, 50, 50, 51, 52, 53, 55, 58, 60, 61, 65, 65, 65, 65, 68, 70, 73, 74, 75, 75, 78, 80, 82, 85, 85, 85, 85, 87, 89, 90, 91, 95, 97, 100, 100, 101, 102, 104, 105, 106, 109, 110, 111, 113, 115, 116, 117, 119, 120

Offset: 1

David W. Wilson

The largest member 'c' of the Pythagorean triples (a,b,c) ordered by increasing c.
If c^2 = a^2 + b^2 (a < b < c) then c^2 = (n^2 + m^2)/2 with n = b - a, m = b + a. - Zak Seidov, Mar 03 2011
Numbers n such that A083025(n) > 0, i.e., n is divisible by at least one prime of the form 4k+1. - Max Alekseyev, Oct 24 2008
A number appears only once in the sequence if and only if it is divisible by exactly one prime of the form 4k+1 with multiplicity one (cf. A084645). - Jean-Christophe Hervé, Nov 11 2013
If c^2 = a^2 + b^2 with a and b > 0, then a <> b: the sum of 2 equal squares cannot be a square because sqrt(2) is not rational. - Jean-Christophe Hervé, Nov 11 2013

W. L. Schaaf, Recreational Mathematics, A Guide To The Literature, "The Pythagorean Relationship", Chapter 6 pp. 89-99 NCTM VA 1963.
W. L. Schaaf, A Bibliography of Recreational Mathematics, Vol. 2, "The Pythagorean Relation", Chapter 6 pp. 108-113 NCTM VA 1972.
W. L. Schaaf, A Bibliography of Recreational Mathematics, Vol. 3, "Pythagorean Recreations", Chapter 6 pp. 62-6 NCTM VA 1973.

Zak Seidov and T. D. Noe, Table of n, a(n) for n = 1..10000 (Zak Seidov entered the first 1981 terms).
Anonymous, Links to Pythagorean Theorem Proofs
Henry Bottomley, Pythagoras's theorem (animated proof)
Kangourou Math Website, L'animation du theoreme de Pythagore
Ron Knott, Pythagorean Triples and Online Calculators
Ron Knott, Right-angled Triangles and Pythagoras' Theorem
I. Kobayashi et al., Pythagorean Theorem (Java Interactive Proofs, Applications and Explorations)
Mathematical Database, Poster, 7 Ways to prove the Pythagorean Theorem
B. Richmond, The Pythagorean Theorem
M. Shepperd, Web Resources on Pythagoras' Theorem
J. S. Silverman, A Friendly Introduction to Number Theory, Chapters 1 to 6 (see Chapters 2 and 3).
G. Villemin's Almanach of Numbers, Triangles & Triplets de Pythagore
Eric Weisstein's World of Mathematics, Pythagorean Triple
Index entries for sequences related to sums of squares

Cf. A009012, A009003, A024507, A004431, A046083, A046084, A004144, A083025, A084645, A084646, A084647, A084648, A084649, A006339.

A009000:=proc(N) # To get all terms <= N
    local p,q,i,L;
    L:=[];
    for p from 2 to floor(sqrt(N-1)) do
        for q to p-1 do
            if igcd(p,q)=1 and is(p-q,odd) then
                L:=[op(L),seq(i*(p^2+q^2),i=1..N/(p^2+q^2))];
            fi
        od
    od;
    return op(sort(L))
end proc:
A009000(120); # Felix Huber, Feb 10 2025

max = 120; hypotenuseQ[n_] := For[k = 1, True, k++, p = Prime[k]; Which[Mod[p, 4] == 1 && Divisible[n, p], Return[True], p > n, Return[False]]]; hypotenuses = Select[Range[max], hypotenuseQ]; red[c_] := {a, b, c} /. {ToRules[ Reduce[0 < a <= b && a^2 + b^2 == c^2, {a, b}, Integers]]}; A009000 = Flatten[red /@ hypotenuses, 1][[All, -1]] (* Jean-François Alcover, May 23 2012, after Max Alekseyev *)
Sqrt[#]&/@Flatten[Table[Total/@Select[IntegerPartitions[n^2,{2}],Length[Union[#]]==2&&AllTrue[Sqrt[#],IntegerQ]&],{n,150}]] (* Harvey P. Dale, May 25 2025 *)

list(lim)=my(v=List(),m2,s2,h2,h); for(middle=4,lim-1, m2=middle^2; for(small=1,middle, s2=small^2; if(issquare(h2=m2+s2,&h), if(h>lim, break); listput(v, h)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 23 2017

list(lim) = {my(lh = List()); for(u = 2, sqrtint(lim), for(v = 1, u, if (u^2+v^2 > lim, break); if ((gcd(u,v) == 1) && (0 != (u-v)%2), for (i = 1, lim, if (i*(u^2+v^2) > lim, break); /* if (u^2 - v^2 < 2*u*v, w = [i*(u^2 - v^2), i*2*u*v, i*(u^2+v^2)], w = [i*2*u*v, i*(u^2 - v^2), i*(u^2+v^2)]); */ listput(lh, i*(u^2+v^2)););););); vecsort(Vec(lh));} \\ Michel Marcus, Apr 10 2021

from math import isqrt
def aupto(limit):
  s = [i*i for i in range(1, limit+1)]
  s2 = sorted(a+b for i, a in enumerate(s) for b in s[i+1:])
  return [isqrt(k) for k in s2 if k in s]
print(aupto(120)) # Michael S. Branicky, May 10 2021

A063725 Number of ordered pairs (x,y) of positive integers such that x^2 + y^2 = n.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Aug 23 2001

nonn

a(A018825(n))=0; a(A000404(n))>0; a(A081324(n))=1; a(A004431(n))>1. - Reinhard Zumkeller, Aug 16 2011

			a(5) = 2 from the solutions (1,2) and (2,1).

Cf. A000404 (the numbers n that can be represented in this form).
Cf. A000161, A063691, A063730, A025426, A000290, A010052.
Column k=2 of A337165.

a063725 n =
   sum $ map (a010052 . (n -)) $ takeWhile (< n) $ tail a000290_list
a063725_list = map a063725 [0..]
-- Reinhard Zumkeller, Aug 16 2011

nn = 100; t = Table[0, {nn}]; s = Sqrt[nn]; Do[n = x^2 + y^2; If[n <= nn, t[[n]]++], {x, s}, {y, s}]; Join[{0}, t] (* T. D. Noe, Apr 03 2011 *)

a(n)=if(n==0, return(0)); my(f=factor(n)); prod(i=1, #f~, if(f[i, 1]%4==1, f[i, 2]+1, f[i, 2]%2==0 || f[i, 1]==2)) - issquare(n) \\ Charles R Greathouse IV, May 18 2016

from math import prod
from sympy import factorint
def A063725(n):
    f = factorint(n)
    return prod(1 if p==2 else (e+1 if p&3==1 else (e+1)&1) for p, e in f.items())-(not any(e&1 for e in f.values())) if n else 0 # Chai Wah Wu, May 17 2023

G.f.: (Sum_{m=1..inf} x^(m^2))^2.
a(n) = ( A004018(n) - 2*A000122(n) + A000007(n) )/4. - Max Alekseyev, Sep 29 2012
G.f.: (theta_3(q) - 1)^2/4, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 08 2018

A003995 Sum of (any number of) distinct squares: of form r^2 + s^2 + t^2 + ... with 0 <= r < s < t < ...

Original entry on oeis.org

0, 1, 4, 5, 9, 10, 13, 14, 16, 17, 20, 21, 25, 26, 29, 30, 34, 35, 36, 37, 38, 39, 40, 41, 42, 45, 46, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 93, 94, 95, 97

Offset: 1

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sums of squares
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A001983, A033461, A008935. Complement of A001422.

Cf. A000290; subsequences: A004431, A004432, A004433, A004434, A224981, A224982, A224983.

a003995 n = a003995_list !! (n-1)
a003995_list = filter (p a000290_list) [0..]
   where p (q:qs) m = m == 0 || q <= m && (p qs (m - q) || p qs m)
-- Reinhard Zumkeller, Apr 22 2013

lim = 10; s = {0}; Do[s = Union[s, s + n^2], {n, lim}]; Select[s, 0 <= # <= lim^2 &] (* T. D. Noe, Jul 10 2012 *)

a(n)=if(n<1,0,n=a(n-1); until(polcoeff(prod(k=1,sqrt(n),1+x^k^2), n), n++); n)

Exponents in expansion of (1+x)*(1+x^4)*(1+x^9)*(1+x^16)*(1+x^25)*(1+x^36)*(1+x^49)*(1+x^64)*(1+x^81)*(1+x^100)*(1+x^121)*(1+x^144)*...

For n > 98, a(n) = n + 30. - Charles R Greathouse IV, Sep 02 2011 (This implies a(n+2) = 2*a(n+1)-a(n) for n > 98.)

A267099 Fully multiplicative involution swapping the positions of 4k+1 and 4k+3 primes: a(1) = 1; a(prime(k)) = A267101(k), a(xy) = a(x)a(y) for x, y > 1.

Original entry on oeis.org

1, 2, 5, 4, 3, 10, 13, 8, 25, 6, 17, 20, 7, 26, 15, 16, 11, 50, 29, 12, 65, 34, 37, 40, 9, 14, 125, 52, 19, 30, 41, 32, 85, 22, 39, 100, 23, 58, 35, 24, 31, 130, 53, 68, 75, 74, 61, 80, 169, 18, 55, 28, 43, 250, 51, 104, 145, 38, 73, 60, 47, 82, 325, 64, 21, 170, 89, 44, 185, 78, 97, 200, 59, 46, 45, 116, 221, 70, 101

Offset: 1

Antti Karttunen, Feb 01 2016

Lexicographically earliest self-inverse permutation of natural numbers where each prime of the form 4k+1 is replaced by a prime of the form 4k+3 and vice versa, with the composite numbers determined by multiplicativity.
Fully multiplicative with a(p_n) = p_{A267100(n)} = A267101(n).
Maps each term of A004613 to some term of A004614, each (nonzero) term of A001481 to some term of A268377 and each term of A004431 to some term of A268378 and vice versa.
Sequences A072202 and A078613 are closed with respect to this permutation.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A000035, A000040, A000720, A010051, A020639, A032742, A267100, A267101, A354102 (Möbius transform), A354103 (inverse Möbius transform), A354192 (fixed points).
Cf. A002144, A002145, A005094, A065338, A072202, A078613, A079635.
Cf. A001481, A268377, A004431, A268378, A004613, A004614.
Cf. also A108548.

up_to = 2^16;
A267097list(up_to) = { my(v=vector(up_to),i=0,c=0); forprime(p=2,prime(up_to), if(1==(p%4), c++); i++; v[i] = c); (v); };
v267097 = A267097list(up_to);
A267097(n) = v267097[n];
A267098(n) = ((n-1)-A267097(n));
list_primes_of_the_form(up_to,m,k) = { my(v=vector(up_to),i=0); forprime(p=2,, if(k==(p%m), i++; v[i] = p; if(i==up_to,return(v)))); };
v002144 = list_primes_of_the_form(2*up_to,4,1);
A002144(n) = v002144[n];
v002145 = list_primes_of_the_form(2*up_to,4,3);
A002145(n) = v002145[n];
A267101(n) = if(1==n,2,if(1==(prime(n)%4),A002145(A267097(n)),A002144(A267098(n))));
A267099(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A267101(primepi(f[k,1]))); factorback(f); }; \\ Antti Karttunen, May 18 2022
(Scheme, with memoization-macro definec)
(definec (A267099 n) (cond ((<= n 1) n) ((= 1 (A010051 n)) (A267101 (A000720 n))) (else (* (A267099 (A020639 n)) (A267099 (A032742 n))))))

a(1) = 1; after which, if n is k-th prime [= A000040(k)], then a(n) = A267101(k), otherwise a(A020639(n)) * a(A032742(n)).
Other identities. For all n >= 1:
a(A000040(n)) = A267101(n).
a(2*n) = 2*a(n).
a(3*n) = 5*a(n).
a(5*n) = 3*a(n).
a(7*n) = 13*a(n).
a(11*n) = 17*a(n).
etc. See examples in A267101.
A000035(n) = A000035(a(n)). [Preserves the parity of n.]
A005094(a(n)) = -A005094(n).
A079635(a(n)) = -A079635(n).

Verbal description prefixed to the name by Antti Karttunen, May 19 2022

A004432 Numbers that are the sum of 3 distinct nonzero squares.

Original entry on oeis.org

14, 21, 26, 29, 30, 35, 38, 41, 42, 45, 46, 49, 50, 53, 54, 56, 59, 61, 62, 65, 66, 69, 70, 74, 75, 77, 78, 81, 83, 84, 86, 89, 90, 91, 93, 94, 98, 101, 104, 105, 106, 107, 109, 110, 113, 114, 115, 116, 117, 118, 120, 121, 122, 125, 126, 129, 131, 133

Offset: 1

N. J. A. Sloane

Numbers that can be written as a(n) = x^2 + y^2 + z^2 with 0 < x < y < z.
This is a subsequence (equal to the range) of A024803. As a set, it is the union of A025339 and A024804, subsequences of numbers having exactly one, resp. more than one, such representations. - M. F. Hasler, Jan 25 2013
Conjecture: a number n is a sum of 3 squares, but not a sum of 3 distinct nonzero squares (i.e., is in A004432 but not A000408), if and only if it is of the form 4^j*s, where j >= 0 and s in {1, 2, 3, 5, 6, 9, 10, 11, 13, 17, 18, 19, 22, 25, 27, 33, 34, 37, 43, 51, 57, 58, 67, 73, 82, 85, 97, 99, 102, 123, 130, 163, 177, 187, 193, 267, 627, 697}. - Jeffrey Shallit, Jan 15 2017
4*a(n) gives the sums of 3 distinct nonzero even squares. - Wesley Ivan Hurt, Apr 05 2021

			14 = 1^2 + 2^2 + 3^2;
62 = 1^2 + 5^2 + 6^2.

Cf. A001974, A024803, A025339, A025442.

Cf. A000290, A003995, A004431, A004433, A004434, A224981, A224982, A224983.

a004432 n = a004432_list !! (n-1)
a004432_list = filter (p 3 $ tail a000290_list) [1..] where
   p k (q:qs) m = k == 0 && m == 0 ||
                  q <= m && k >= 0 && (p (k - 1) qs (m - q) || p k qs m)
-- Reinhard Zumkeller, Apr 22 2013

f[upto_]:=Module[{max=Floor[Sqrt[upto]]},Select[Union[Total/@ (Subsets[ Range[ max],{3}]^2)],#<=upto&]]; f[150]  (* Harvey P. Dale, Mar 24 2011 *)

is_A004432(n)=for(x=1,sqrtint(n\3),for(y=x+1,sqrtint((n-1-x^2)\2),issquare(n-x^2-y^2)&return(1)))  \\ M. F. Hasler, Feb 02 2013

A004432 = { x^2 + y^2 + z^2; 0 < x < y < z }.

n is in A004432 <=> A025442(n) > 0. - M. F. Hasler, Feb 03 2013

A025441 Number of partitions of n into 2 distinct nonzero squares.

Original entry on oeis.org

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

Offset: 0

David W. Wilson

nonn

T. D. Noe, Table of n, a(n) for n = 0..10000
Michael Gilleland, Some Self-Similar Integer Sequences
Index entries for sequences related to sums of squares

Cf. A060306 gives records; A052199 gives where records occur.
Cf. A000161, A000290, A010052, A025435, A157228, A053866, A145393, A093709.
Column k=2 of A341040.
Cf. A004439 (a(n)=0), A025302 (a(n)=1), A025303 (a(n)=2), A025304 (a(n)=3), A025305 (a(n)=4), A025306 (a(n)=5), A025307 (a(n)=6), A025308 (a(n)=7), A025309 (a(n)=8), A025310 (a(n)=9), A025311 (a(n)=10), A004431 (a(n)>0).

a025441 n = sum $ map (a010052 . (n -)) $
                      takeWhile (< n `div` 2) $ tail a000290_list
-- Reinhard Zumkeller, Dec 20 2013

Table[Count[PowersRepresentations[n, 2, 2], pr_ /; Unequal @@ pr && FreeQ[pr, 0]], {n, 0, 107}] (* Jean-François Alcover, Mar 01 2019 *)

a(n)=if(n>4,sum(k=1,sqrtint((n-1)\2),issquare(n-k^2)),0) \\ Charles R Greathouse IV, Jun 10 2016

a(n)=if(n<5,return(0)); my(v=valuation(n, 2), f=factor(n>>v), t=1); for(i=1, #f[, 1], if(f[i, 1]%4==1, t*=f[i, 2]+1, if(f[i, 2]%2, return(0)))); if(t%2, t-(-1)^v, t)/2-issquare(n/2) \\ Charles R Greathouse IV, Jun 10 2016

from math import prod
from sympy import factorint
def A025441(n):
    f = factorint(n).items()
    return -int(not (any((e-1 if p == 2 else e)&1 for p,e in f) or n&1)) + (((m:=prod(1 if p==2 else (e+1 if p&3==1 else (e+1)&1) for p, e in f))+((((~n & n-1).bit_length()&1)<<1)-1 if m&1 else 0))>>1) if n else 0 # Chai Wah Wu, Sep 08 2022

a(A025302(n)) = 1. - Reinhard Zumkeller, Dec 20 2013
a(n) = Sum_{ m: m^2|n } A157228(n/m^2). - Andrey Zabolotskiy, May 07 2018
a(n) = [x^n y^2] Product_{k>=1} (1 + y*x^(k^2)). - Ilya Gutkovskiy, Apr 22 2019
a(n) = Sum_{i=1..floor((n-1)/2)} c(i) * c(n-i), where c is the square characteristic (A010052). - Wesley Ivan Hurt, Nov 26 2020
a(n) = A000161(n) - A093709(n). - Andrey Zabolotskiy, Apr 12 2022

cp's OEIS Frontend

A000404 Numbers that are the sum of 2 nonzero squares.

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A009003 Hypotenuse numbers (squares are sums of 2 nonzero squares).

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A008846 Hypotenuses of primitive Pythagorean triangles.

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A024670 Numbers that are sums of 2 distinct positive cubes.

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A009000 Ordered hypotenuse numbers (squares are sums of 2 distinct nonzero squares).

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A267099 Fully multiplicative involution swapping the positions of 4k+1 and 4k+3 primes: a(1) = 1; a(prime(k)) = A267101(k), a(xy) = a(x)a(y) for x, y > 1.