A281229 - OEIS

%I #47 Dec 30 2020 17:10:55

%S 1,2,5,10,13,26,29,34,41,58,61,74,89,106,113,146,149,194,181,202,233,

%T 274,269,386,313,346,389,394,421,458,521,514,557,586,613,698,709,794,

%U 761,802,853,914,929,1018,1013,1186,1109,1154,1201,1282,1301,1354,1409

%N Smallest number k of the form x^2 + y^2 with 0 <= x <= y such that gcd(x, y) = 1, x + y = n, and k has no other decompositions into a sum of two squares.

%C Conjecture: for each n there exists such a number k.

%C Note: a(2m+1) > 1 is a prime p and a(2m) > 2 is a double prime 2q, where p and q are primes == 1 (mod 4).

%C For odd n > 1, a(n) is the smallest prime of the form x^2 + (n - x)^2.

%C For even n > 2, a(n) is the smallest double prime of the above form.

%H Robert Israel, <a href="/A281229/b281229.txt">Table of n, a(n) for n = 1..10000</a>

%F For m > 0, a(2m+1) = A159351(m).

%F For m > 1, a(2m) = 2 * A068486(m).

%p f:= proc(n) local k,v;

%p   for k from ceil(n/2) to n do

%p     v:= k^2+(n-k)^2;

%p     if n::odd then if isprime(v) then return v fi

%p     elif isprime(v/2) then return v

%p     fi

%p   od;

%p   FAIL

%p end proc:

%p f(1):=1: f(2):= 2:

%p map(f, [$1..100]); # _Robert Israel_, Dec 30 2020

%o (PARI) isok(k, n) = {nba = 0; nbb = 0; for (x=0, k, if (issquare(x) && issquare(k-x), if (x <= k - x, nba++; if (nba > 1, return (0)); rx = sqrtint(x); ry = sqrtint(k-x); if ((gcd(rx,ry)==1) && (rx+ry == n), nbb++;);););); if (nbb, return (k), return(0));}

%o a(n) = {k = 1; while (! (s = isok(k, n)), k++; ); s;} \\ _Michel Marcus_, Jan 20 2017

%Y Cf. A002144, A068486, A159351.

%K nonn

%O 1,2

%A _Thomas Ordowski_, Jan 18 2017

%E More terms from _Altug Alkan_, Jan 18 2017

%E More terms from _Jon E. Schoenfield_, Jan 18 2017