A378945 -id:A378945 - 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.

A378946 Locations of records in A378898.

Original entry on oeis.org

1, 3, 13, 31, 49, 68, 216, 227, 288, 339, 408, 421, 797, 1176, 1494, 1947, 3876, 6453, 12108, 12558, 13272, 24027, 80667, 92472, 98154, 186543, 765351, 2294838, 6815886, 11105034, 12608001, 13669797, 25343472, 25485726, 40937853, 48562668, 72974013, 122175969

Offset: 1

Robert Israel, Dec 11 2024

nonn

The record values are in A378945.

Numbers k such that for some m, (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.

			a(1) = 1, as A378898(1) = 1, with (1+1)^2 + 1^2 = 5 prime.
a(2) = 3, as A378898(3) = 5, with (5+3)^2 + 3^2 = 73 prime, and 3 is the first k with  A378898(k) > 1.
a(3) = 13, as A378898(13) = 7, with (7+13)^2 + 13^2 = 569 prime, and 13 is the first k with A378898(k) > 5.

Cf. A378898, A378945.

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:
J:= NULL: count:= 0: rec:= 0:
for k from 1 while count < 30 do
  v:= f(k);
  if v > rec then
    count:= count+1;
    J:= J, k;
    rec:= v;
  fi
od:
J;

A378898(a(n)) = A378945(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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