OFFSET
1,1
COMMENTS
The next term is only defined if the set S above is not empty.
Conjecture: the sequence is well defined.
a(13) has 1654 digits. If S is not empty, then its maximal element is P(x) where x is the least positive integer x <= p/2 such that P(x) is prime. - Chai Wah Wu, Jan 09 2017
LINKS
Chai Wah Wu, Table of n, a(n) for n = 1..12
EXAMPLE
a(2) = 7, since 7 = max S_3, where S_3 = {x^2-3x+9 : x is an integer with 0<x<2, and x^2-3x+9 is a prime number}. Clearly, S_3={7}, thus a(2)=7. Now we explain why a(3)=43. We have 43 = max S_7. S_7 := {x^2-7x+49 : x is an integer, 0 <x<7, and x^2-7x+49 is a prime number}. By computations S_7 = {37,43}. Thus a(3) = max S_7 = 43. We explain also why a(4) = 1693. One has 1693 = max S_43, where S_43 = {x^2-43x+43^2 : x is an integer, 0 <x < 43, and x^2-43x+43^2 is a prime number}. By computations S_43 = {1399,1429,1459,1543,1597,1627,1693}. Thus a(3) = max S_43 = 1693.
MAPLE
with(numtheory):
xa := proc(aa) local P, x, a, a2, mi, mm; a:= aa; a2 := a^2; mi := 0; for x from 1 to a-1 do P := x^2-a*x+a2; if isprime(P) then mi := max(P, mi); fi; od;; mi; end;
F := proc(n) option remember if n=1 then return(3); fi; if n=2 then xa(3); else xa(F(n-1)); fi; end;
MATHEMATICA
PROG
(Python)
from __future__ import division
from sympy import isprime
A280717_list, n = [3], 3
for _ in range(10):
for i in range(1, n//2+1):
j = i**2+n*(n-i)
if isprime(j):
n = j
A280717_list.append(n)
break # Chai Wah Wu, Jan 09 2017
CROSSREFS
KEYWORD
nonn,hard
AUTHOR
Luis H. Gallardo, Jan 07 2017
EXTENSIONS
a(5) corrected and a(6)-a(9) added by Chai Wah Wu, Jan 09 2017
STATUS
approved