A371024 a(n) is the least prime p such that p + 4*k*(k+1) is prime for 0 <= k <= n-1 but not for k=n.

2, 3, 29, 5, 23, 269, 272879, 149, 61463, 929, 7426253, 2609, 233, 59, 78977932125503

Offset: 1

Author

J.W.L. (Jan) Eerland, Mar 08 2024

Keywords

• nonn
• more

Comments

a(15) > 3277860277, a(16) > 3103623446, a(17) > 2853255995,

a(18) = 653, a(19) > 2480173428, a(20) > 2058783580, a(21) > 1894529774, a(22) > 1896261075, a(23) > 1836831342, a(24), ..., a(100) > 15000000.

Other than a(1)-a(14) and a(18), no terms < 24870000007. - Michael S. Branicky, Apr 12 2024

From David A. Corneth, Apr 12 2024: (Start)

Using remainders mod q we can restrict the search. For example for a(15) a term can only be 2, 3 or 5 (mod 7). Or maybe 7 itself. If a(15) = p == 1 (mod 7) then for k = 3 we have q + 4*3*(3+1) == 0 mod 7. Similarily number 0, 4 and 6 (mod 7) produce a multiple of 7 where they should not.

Doing so for various primes mod q we can reduce the number of remainders and with that the search space by combining the possible remainders using the Chinese Remainder Theorem (CRT).

So the possible remainders mod 2 are 1. The possible remainders mod 3 are 2. Using the CRT, a number of the form 1 (mod 2) and 2 (mod 3) simultaneously is of the form 5 (mod 6).

a(15) > 2.3*10^13 if it exists. (End)

Crossrefs

Cf. A164926, A370387.

Programs

Maple

f:= proc(p) local k;
  for k from 1 while isprime(p+k*(k+1)*4) do od:
  k
end proc:
A:= Vector(12): count:= 0:
for i from 1 while count < 12 do
  v:= f(ithprime(i));
  if A[v] = 0 then count:= count+1; A[v]:= ithprime(i) fi
od:
convert(A,list);

Mathematica

Table[p=1;m=4;Monitor[Parallelize[While[True,If[And[MemberQ[PrimeQ[Table[p+m*k*(k+1),{k,0,n-1}]],False]==False,PrimeQ[p+m*n*(n+1)]==False],Break[]];p++];p],p],{n,1,10}]

PARI

isok(p, n) = for (k=0, n-1, if (! isprime(p + 4*k*(k+1)), return(0))); return (!isprime(p + 4*n*(n+1)));
a(n) = my(p=2); while (!isok(p, n), p=nextprime(p+1)); p; \\ Michel Marcus, Mar 12 2024

Perl

use ntheory qw(:all); sub a { my $n = $[0]; my $lo = 2; my $hi = 2*$lo; while (1) { my @terms = grep { !is_prime($ + 4*$n*($n+1)) } sieve_prime_cluster($lo, $hi, map { 4*$*($+1) } 1 .. $n-1); return $terms[0] if @terms; $lo = $hi+1; $hi = 2*$lo; } }; $| = 1; for my $n (1..100) { print a($n), ", " }; # Daniel Suteu, Dec 17 2024

Python

from sympy import isprime, nextprime
from itertools import count, islice
def f(p):
    k = 1
    while isprime(p+4*k*(k+1)): k += 1
    return k
def agen(verbose=False): # generator of terms
    adict, n, p = dict(), 1, 1
    while True:
        p = nextprime(p)
        v = f(p)
        if v not in adict:
            adict[v] = p
            if verbose: print("FOUND", v, p)
        while n in adict:
            yield adict[n]; n += 1
print(list(islice(agen(), 14))) # Michael S. Branicky, Apr 12 2024

Extensions

a(15) from Daniel Suteu, Dec 17 2024