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A004431 Numbers that are the sum of 2 distinct nonzero squares. 74
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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)
LINKS
EXAMPLE
53 = 7^2 + 2^2, so 53 is in the sequence.
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 *)
PROG
(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
(Haskell)
import Data.List (findIndices)
a004431 n = a004431_list !! (n-1)
a004431_list = findIndices (> 1) a063725_list
-- Reinhard Zumkeller, Aug 16 2011
(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
CROSSREFS
Complement of A004439.
Sequence in context: A242898 A230486 A024507 * A025302 A268379 A221265
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified March 19 06:50 EDT 2024. Contains 370953 sequences. (Running on oeis4.)