1,1

From the formula it is easy to see that if n is in this sequence, then so are all odd powers of n. - 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 n = x^2 + y^2 with x != y, then n^2 = (x^2-y^2)^2 + (2xy)^2 and n^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

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.

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.

Etienne Fouvry, Claude Levesque, 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

Let n = 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 n is a member iff 1) b_j=0 mod 2 for j=1..s and 2) r>0 or t=1 mod 2 (or both).

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.

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 *)

(PARI) 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

(PARI) 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

(Haskell)

import Data.List (findIndices)

a000404 n = a000404_list !! (n-1)

a000404_list = findIndices (> 0) a025426_list

-- Reinhard Zumkeller, Aug 16 2011

(MAGMA) 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

(GAP) 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

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).

Column k=2 of A336725.

Sequence in context: A189365 A024509 A084889 * A025284 A140328 A000415

Adjacent sequences:  A000401 A000402 A000403 * A000405 A000406 A000407

nonn,nice,easy

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

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

approved