A376920 - OEIS

%I #15 Oct 14 2024 15:35:18

%S 2,19,13,53,31,71,109,263,239,167,661,439,673,1289,1021,2531,3617,

%T 5101,1033,3037,2017,4889,4751,3169,887,2521,11467,20143,563,1873,

%U 9931,2617,9833,12739,7057,78787,58067,10831,29759,22229,62801,65479,12163,20233,16561,87911,26597,28621,148339,44159

%N a(n) is the least prime p such that n^2 + (p-n)^2 is prime and k^2 + (p-k)^2 is composite for 1 <= k < n.

%C a(n) is the least prime p such that A260870((p-1)/2) = n.

%H Robert Israel, <a href="/A376920/b376920.txt">Table of n, a(n) for n = 1..220</a>

%e a(3) = 13 because 13 is prime, 3^2 + (13-3)^2 = 109 is prime, and both 1^2 + (13-1)^2 = 145 and 2^2 + (13-2)^2 = 125 are composite, and no smaller prime works.

%p N:= 100: # for a(1) .. a(N)

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

%p      for k from 1 to p/2 do if isprime(k^2 + (p-k)^2) then return k fi od;

%p      FAIL

%p end proc:

%p V:= Vector(N): count:= 0: p:= 0:

%p for i from 1 while count < N do

%p   p:= nextprime(p);

%p   v:= fp(p);

%p   if v <= N and V[v] = 0 then V[v]:= p; count:= count+1 fi

%p od:

%p convert(V,list);

%o (Python)

%o from sympy import isprime, nextprime

%o def A376920(n):

%o     p = n

%o     while (p:=nextprime(p)):

%o         if isprime(n**2+(p-n)**2) and not any(isprime(k**2+(p-k)**2) for k in range(1,min(n-1,p//2)+1)):

%o             return p # _Chai Wah Wu_, Oct 14 2024

%Y Cf. A260869, A260870.

%K nonn

%O 1,1

%A _Robert Israel_, Oct 10 2024