A378946 - OEIS

%I #8 Dec 12 2024 09:28:03

%S 1,3,13,31,49,68,216,227,288,339,408,421,797,1176,1494,1947,3876,6453,

%T 12108,12558,13272,24027,80667,92472,98154,186543,765351,2294838,

%U 6815886,11105034,12608001,13669797,25343472,25485726,40937853,48562668,72974013,122175969

%N Locations of records in A378898.

%C The record values are in A378945.

%C 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.

%F A378898(a(n)) = A378945(n).

%e a(1) = 1, as A378898(1) = 1, with (1+1)^2 + 1^2 = 5 prime.

%e 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.

%e 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.

%p f:= proc(k) local m;

%p   for m from 1 by 2 do

%p     if igcd(m,k) = 1 and isprime((k+m)^2 + k^2) then return m fi

%p   od

%p end proc:

%p J:= NULL: count:= 0: rec:= 0:

%p for k from 1 while count < 30 do

%p   v:= f(k);

%p   if v > rec then

%p     count:= count+1;

%p     J:= J, k;

%p     rec:= v;

%p   fi

%p od:

%p J;

%Y Cf. A378898, A378945.

%K nonn

%O 1,2

%A _Robert Israel_, Dec 11 2024