cp's OEIS Frontend

A383595 a(n) is the smallest prime k such that (prime(n), k, u, v) are the vertices of a square in Ulam's spiral, where k < u < v are all primes; or -1 if there is no such k.

%I A383595 #13 May 07 2025 03:23:30
%S A383595 -1,-1,-1,56527,59,67,251,-1,-1,2473,3001,43,43,41,173,1621,61,59,13,
%T A383595 141937,13,13,10459,331,33211,643,179,41,41,1429,11,11,59,59,13,127,
%U A383595 163,157,169957,47,103,56519,683,2843,6841,211,199,311,59407,439,11,137,274831
%N A383595 a(n) is the smallest prime k such that (prime(n), k, u, v) are the vertices of a square in Ulam's spiral, where k < u < v are all primes; or -1 if there is no such k.
%C A383595 For each prime number prime(n) in Ulam's spiral, we search for the least 3 primes k, u, v, with k < u < v, such that (prime(n), k, u, v) are the vertices of a square whose sides are parallel to the rows and columns of the spiral, where a(n) equals k.
%C A383595 Given a prime p, some vertices are close to p. For example, for prime 13, the vertices are (13, 67, 73, 79), while for others they are not, such as 7, where the least primes are (7, 56527, 58567, 58687). On the other hand, primes such as 2, 3, 5, 19 and 23 are not vertices of any square with prime vertices.
%C A383595 Conjecture: if a prime is vertex of a square of prime vertices, then it is vertex of infinitely many squares whose vertices are prime. For example, in the case of 11, some of them are: (11, 59, 127, 131), (11, 137, 233, 239), (11, 769, 977, 991).
%C A383595 Questions: Which prime numbers are not vertices of any square with prime vertices? What condition must they satisfy?
%C A383595 Are there infinite primes p and q which are vertices of two squares with prime vertices? For example, 47 and 353 are vertices of the squares (47, 353, 109, 347) and (47, 353, 173, 359).
%e A383595 For A000040(5) = 11, it is observed that 11 together with 127, 131 and 59 are the vertices of a square whose center is 55. And this is the smallest square of prime vertices that has 11 as one of its vertices. Since 59 is the smallest number between 127, 131 and 59, then a(5) = 59.
%e A383595     .  .  .  .  .
%e A383595   —11-28-53-86-127—
%e A383595   —12-29-54-87-128—
%e A383595   —13-30-55-88-129—
%e A383595   —32-31-56-89-130—
%e A383595   —59-58-57-90-131—
%e A383595     .  .  .  .  .
%Y A383595 Cf. A063826, A000040.
%K A383595 sign
%O A383595 1,4
%A A383595 _Gonzalo Martínez_, May 01 2025