A004431 Numbers that are the sum of 2 distinct nonzero squares.

5, 10, 13, 17, 20, 25, 26, 29, 34, 37, 40, 41, 45, 50, 52, 53, 58, 61, 65, 68, 73, 74, 80, 82, 85, 89, 90, 97, 100, 101, 104, 106, 109, 113, 116, 117, 122, 125, 130, 136, 137, 145, 146, 148, 149, 153, 157, 160, 164, 169, 170, 173, 178, 180, 181, 185, 193, 194, 197

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Numbers whose prime factorization includes at least one prime congruent to 1 mod 4 and any prime factor congruent to 3 mod 4 has even multiplicity. - Franklin T. Adams-Watters, May 03 2006

Reordering of A055096 by increasing values and without repetition. - Paul Curtz, Sep 08 2008

A063725(a(n)) > 1. - Reinhard Zumkeller, Aug 16 2011

The square of these numbers is also the sum of two nonzero squares, so this sequence is a subsequence of A009003. - Jean-Christophe Hervé, Nov 10 2013

Closed under multiplication. Primitive elements are those with exactly one prime factor congruent to 1 mod 4 with multiplicity one (A230779). - Jean-Christophe Hervé, Nov 10 2013

From Bob Selcoe, Mar 23 2016: (Start)

Numbers c such that there is d < c, d >= 1 where c + d and c - d are square. For example, 53 + 28 = 81, 53 - 28 = 25.

Given a prime p == 1 mod 4, a term appears if and only if it is of the form p^i, p*2^j or p*k^2 {i,j,k >= 1}, or a product of any combination of these forms. Therefore, the products of any terms to any powers also are terms. For example, p(1) = 5 and p(2) = 13 so term 45 appears because 5*3^2 = 45 and term 416 appears because 13*2^5 = 416; therefore 45 * 416 = 18720 appears, as does 45^3 * 416^11 = 18720^3 * 416^8.

Numbers of the form j^2 + 2*j*k + 2*k^2 {j,k >= 1}. (End)

Suppose we have a term t = x^2 + y^2. Then s^2*t = (s*x)^2 + (s*y)^2 is a term for any s > 0. Also 2*t = (y+x)^2 + (x-y)^2 is a term. It follows that q*s^2*t is a term for any s > 0 and q=1 or 2. Examples: 2*7^2*26 = 28^2 + 42^2; 6^2*17 = 6^2 + 24^2. - Jerzy R Borysowicz, Aug 11 2017

To find terms up to some upper bound u, we can search for x^2 + y^2 = t where x is odd and y is even. Then we add all numbers of the form 2^m * t <= u and then remove duplicates. - David A. Corneth, Oct 04 2017

From Bernard Schott, Apr 13 2022: (Start)

The 5th comment "Closed under multiplication" can be proved with Brahmagupta's identity: (a^2+b^2) * (c^2+d^2) = (ac + bd)^2 + (ad - bc)^2.

The subsequence of primes is A002144. (End)

Examples

			53 = 7^2 + 2^2, so 53 is in the sequence.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000.
• Wikipedia, Brahmagupta's identity.
• Index entries for sequences related to sums of squares.

Crossrefs

Complement of A004439.

Cf. A000404, A002144, A007692, A009000, A009003, A009177, A024507, A081324, A081325, A118882, A230779.

Programs

Haskell

import Data.List (findIndices)
a004431 n = a004431_list !! (n-1)
a004431_list = findIndices (> 1) a063725_list
-- Reinhard Zumkeller, Aug 16 2011

Maple

isA004431 := proc(n)
    local a,b ;
    for a from 2 do
        if a^2>= n then
            return false;
        end if;
        b := n -a^2 ;
        if b < 1 then
            return false ;
        end if;
        if issqr(b) then
            if ( sqrt(b) <> a ) then
                return true;
            end if;
        end if;
    end do:
    return false;
end proc:
A004431 := proc(n)
    option remember ;
    local a;
    if n = 1 then
        5;
    else
        for a from procname(n-1)+1 do
            if isA004431(a) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Jan 29 2013

Mathematica

A004431 = {}; Do[a = 2 m * n; b = m^2 - n^2; c = m^2 + n^2; AppendTo[A004431, c], {m, 100}, {n, m - 1}]; Take[Union@A004431, 63] (* Robert G. Wilson v, May 02 2009 *)
Select[Range@ 200, Length[PowersRepresentations[#, 2, 2] /. {{0, } -> Nothing, {a, b_} /; a == b -> Nothing}] > 0 &] (* Michael De Vlieger, Mar 24 2016 *)

PARI

select( isA004431(n)={n>1 && vecmin((n=factor(n)%4)[,1])==1 && ![f[1]>2 && f[2]%2 | f<-n~]}, [1..199]) \\ M. F. Hasler, Feb 06 2009, updated Nov 24 2019

PARI

is(n)=if(n<5, return(0)); my(f=factor(n)%4); if(vecmin(f[, 1])>1, return(0)); for(i=1, #f[, 1], if(f[i, 1]==3 && f[i, 2]%2, return(0))); 1
for(n=1, 1e3, if(is(n), print1(n, ", "))) \\ Altug Alkan, Dec 06 2015

PARI

upto(n) = {my(res = List(), s); forstep(i=1, sqrtint(n), 2, forstep(j = 2, sqrtint(n - i^2), 2, listput(res, i^2 + j^2))); s = #res; for(i = 1, s, t = res[i]; for(e = 1, logint(n \ res[i], 2), listput(res, t<<=1))); listsort(res, 1); res} \\ David A. Corneth, Oct 04 2017

Python

def aupto(limit):
  s = [i*i for i in range(1, int(limit**.5)+2) if i*i < limit]
  s2 = set(a+b for i, a in enumerate(s) for b in s[i+1:] if a+b <= limit)
  return sorted(s2)
print(aupto(197)) # Michael S. Branicky, May 10 2021