A281229 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.

1, 2, 5, 10, 13, 26, 29, 34, 41, 58, 61, 74, 89, 106, 113, 146, 149, 194, 181, 202, 233, 274, 269, 386, 313, 346, 389, 394, 421, 458, 521, 514, 557, 586, 613, 698, 709, 794, 761, 802, 853, 914, 929, 1018, 1013, 1186, 1109, 1154, 1201, 1282, 1301, 1354, 1409

Offset: 1

Author

Thomas Ordowski, Jan 18 2017

Keywords

• nonn

Comments

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

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).

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

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

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002144, A068486, A159351.

Programs

Maple

f:= proc(n) local k,v;
  for k from ceil(n/2) to n do
    v:= k^2+(n-k)^2;
    if n::odd then if isprime(v) then return v fi
    elif isprime(v/2) then return v
    fi
  od;
  FAIL
end proc:
f(1):=1: f(2):= 2:
map(f, [$1..100]); # Robert Israel, Dec 30 2020

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));}
a(n) = {k = 1; while (! (s = isok(k, n)), k++; ); s;} \\ Michel Marcus, Jan 20 2017

Formula

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

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

Extensions

More terms from Altug Alkan, Jan 18 2017

More terms from Jon E. Schoenfield, Jan 18 2017