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

1,2

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

Formula

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

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

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

Prog

(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

Crossrefs

Cf. A002144, A068486, A159351.

Sequence in context: A064233 A051952 A103188 * A185647 A064392 A328700

Adjacent sequences: A281226 A281227 A281228 * A281230 A281231 A281232

Keyword

nonn

Author

Thomas Ordowski, Jan 18 2017

Extensions

More terms from Altug Alkan, Jan 18 2017

More terms from Jon E. Schoenfield, Jan 18 2017

Status

approved