A378946 -id:A378946 - cp's OEIS frontend

A378898 a(n) is the least k > 0 such that (n+k)^2 + n^2 is prime.

1, 1, 5, 1, 1, 5, 1, 5, 1, 3, 3, 1, 7, 1, 7, 3, 1, 5, 1, 3, 5, 1, 7, 1, 1, 5, 5, 5, 1, 1, 13, 1, 7, 1, 1, 13, 3, 7, 1, 3, 3, 1, 5, 5, 7, 3, 1, 5, 25, 1, 5, 5, 5, 5, 3, 5, 11, 5, 5, 1, 3, 3, 17, 7, 1, 5, 13, 27, 1, 1, 13, 1, 27, 5, 19, 9, 3, 5, 1, 9, 19, 1, 5, 1, 1, 9, 1, 15, 7, 1, 3, 3, 5, 5, 7

Offset: 1

Robert Israel, Dec 11 2024

nonn

			a(3) = 5 because (3+5)^2 + 3^2 = 73 is prime, and no smaller number works.

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

Cf. A027861 (a(n) = 1), A089489, A378945, A378946.

f:= proc(n) local k;
  for k from n+1 by 2 do
    if igcd(k,n) = 1 and isprime(k^2 + n^2) then return k-n fi
  od
end proc;
map(f, [$1..100]);

a(n) = my(k=1); while (!isprime((n+k)^2 + n^2), k++); k; \\ Michel Marcus, Dec 11 2024

a(n) = A089489(n) - n.

A378945 Record values in A378898.

Original entry on oeis.org

1, 5, 7, 13, 25, 27, 29, 31, 35, 41, 47, 53, 65, 73, 77, 103, 113, 119, 149, 179, 181, 215, 233, 235, 251, 319, 413, 425, 433, 455, 473, 485, 491, 529, 535, 557, 659, 725

Offset: 1

Robert Israel, Dec 11 2024

Numbers m > 0 such that for some k, (m+k)^2 + k^2 is prime while (m'+k)^2 + k^2 is not prime for 0 < m' < m, and for every k' < k there is m' < m such that (m'+k')^2 + k'^2 is prime.

The values of k are in A378946.

			a(1) = 1 = A378898(1), as (1+1)^2 + 1^2 = 5 is prime.
a(2) = 5 = A378898(3), as (5+3)^2 + 3^2 = 73 is prime, is the first value of A378898 greater than 1.
a(3) = 7 = A378898(13), as (7+13)^2 + 13^2 = 569 is prime, is the first value of A378898 greater than 5.

Cf. A378898, A378946.

f:= proc(k) local m;
  for m from 1 by 2 do
    if igcd(m,k) = 1 and isprime((k+m)^2 + k^2) then return m fi
  od
end proc:
R:= NULL: count:= 0: rec:= 0:
for k from 1 while count < 30 do
  v:= f(k);
  if v > rec then
    count:= count+1;
    R:= R, v;
    rec:= v;
  fi
od:
R;

a(n) = A378898(A378946(n)).

cp's OEIS Frontend

A378898 a(n) is the least k > 0 such that (n+k)^2 + n^2 is prime.

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