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

Offset

1,1

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)

Links

Table of n, a(n) for n=1..14.

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}]

Prog

(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
(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

Crossrefs

Cf. A164926, A370387.

Sequence in context: A206591 A003017 A096580 * A351693 A324941 A028868

Adjacent sequences: A371021 A371022 A371023 * A371025 A371026 A371027

Keyword

nonn,more

Author

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

Status

approved