A068486 -id:A068486 - cp's OEIS frontend

A069003 Smallest integer d such that n^2 + d^2 is a prime number.

1, 1, 2, 1, 2, 1, 2, 3, 4, 1, 4, 7, 2, 1, 2, 1, 2, 5, 6, 1, 4, 5, 8, 1, 4, 1, 2, 5, 4, 11, 4, 3, 2, 5, 2, 1, 2, 3, 10, 1, 4, 5, 8, 9, 2, 5, 2, 13, 4, 7, 4, 3, 10, 1, 4, 1, 2, 3, 6, 13, 10, 3, 32, 9, 2, 1, 2, 5, 10, 3, 6, 5, 2, 1, 4, 5, 10, 7, 4, 7, 4, 3, 18, 1, 2, 9, 2, 3, 4, 1, 4, 7, 8, 1, 2, 5, 2, 3, 4, 3

Offset: 1

T. D. Noe, Apr 02 2002

With i being the imaginary unit, n + di is the smallest Gaussian prime with real part n and a positive imaginary part. Likewise for n - di. See A002145 for Gaussian primes with imaginary part 0. - Alonso del Arte, Feb 07 2011
Conjecture: a(n) does not exceed 4*sqrt(n+1) for any positive integer n. - Zhi-Wei Sun, Apr 15 2013
Conjecture holds for the first 15*10^6 terms. - Joerg Arndt, Aug 19 2014
Infinitely many d exist such that n^2 + d^2 is prime, under Schinzel's Hypothesis H; see Sierpinski (1988), p. 221. - Jonathan Sondow, Nov 09 2015

			a(5)=2 because 2 is the smallest integer d such that 5^2+d^2 is a prime number.

W. Sierpinski, Elementary Theory of Numbers, 2nd English edition, revised and enlarged by A. Schinzel, Elsevier, 1988.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Gaussian Prime.
Wikipedia, Schinzel's Hypothesis H.

Cf. A068486 (lists the prime numbers n^2 + d^2).
Cf. A185636, A204065.
Cf. A239388, A239389 (record values).
Cf. A053000.

f:= proc(n) local d;
     for d from 1+(n mod 2) by 2 do
       if isprime(n^2+d^2) then return d fi
     od
end proc:
f(1):= 1:
map(f, [$1..1000]); # Robert Israel, Jul 06 2015

imP4P[n_] := Module[{k = 1}, While[Not[PrimeQ[n^2 + k^2]], k++]; k]; Table[imP4P[n], {n, 50}] (* Alonso del Arte, Feb 07 2011 *)

a(n)=my(k);while(!isprime(n^2+k++^2),);k \\ Charles R Greathouse IV, Mar 20 2013

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.

Original entry on oeis.org

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

Thomas Ordowski, Jan 18 2017

nonn

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.

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

Cf. A002144, A068486, A159351.

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

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

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

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

More terms from Altug Alkan, Jan 18 2017

More terms from Jon E. Schoenfield, Jan 18 2017

cp's OEIS Frontend

A069003 Smallest integer d such that n^2 + d^2 is a prime number.

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