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A004431
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Numbers that are the sum of 2 distinct nonzero squares.
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74
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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)
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OFFSET
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1,1
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COMMENTS
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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
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
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
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)
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LINKS
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EXAMPLE
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53 = 7^2 + 2^2, so 53 is in the sequence.
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MAPLE
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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:
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;
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MATHEMATICA
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Select[Range@ 200, Length[PowersRepresentations[#, 2, 2] /. {{0, _} -> Nothing, {a_, b_} /; a == b -> Nothing}] > 0 &] (* Michael De Vlieger, Mar 24 2016 *)
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PROG
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(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
(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)
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CROSSREFS
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Cf. A000404, A002144, A007692, A009000, A009003, A009177, A024507, A081324, A081325, A118882, A230779.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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