A385077 Primes x that cannot be written as x = p + 2*q where p < q are primes.

2, 3, 5, 7, 11, 23, 83, 167

Offset

1,1

Comments

Also, primes x with no strict-order Lemoine representation x = p + 2*q (primes p < q).

Conjectured to be finite (see also conjectures in A046927, A337924). Note that Lemoine representations (no restriction on order of p, q) hold for all odd numbers up to 10^13 (see Juhász et al.).

a(9), if its exists, exceeds 10^8.

a(9), if its exists, exceeds 10^16. - Martin Ehrenstein, Aug 10 2025

Links

Table of n, a(n) for n=1..8.

Zsófia Juhász, Máté Bartalos, Péter Magyar, and Gábor Farkas, Empirical verification of a new generalisation of Goldbach's conjecture up to 10^12 (or 10^13) for all coefficients <= 40, arXiv:2304.00024 [math.GM], 2023.

Example

13 = 3 + 2*5 with 3 < 5, so 13 is not a term.
No such representation exists for 23, so 23 is a term.

Mathematica

lim=200; p=Prime[Range[PrimePi[lim/2]]]; s={}; Do[AppendTo[s, p[[i]]+2*p[[j]]], {i, PrimePi[lim/2]}, {j, i+1, PrimePi[lim/2]}]; s=Select[Union[s], PrimeQ[#]&]; Complement[Prime[Range[PrimePi[lim]]], s] (* James C. McMahon, Jul 28 2025 *)

Prog

(Python)
from sympy import primerange, isprime
def aupto(limit=10000):
    P = list(primerange(2, limit+1))
    return [p for p in P if all((p-a)&1 or (p-a)//2<=a or not isprime((p-a)//2) for a in P if a<p)]
print(aupto())

Crossrefs

Cf. A046927, A337924.

Sequence in context: A373044 A378774 A036342 * A114421 A387593 A167119

Adjacent sequences: A385074 A385075 A385076 * A385078 A385079 A385080

Keyword

nonn,hard,more

Author

Ian Shannon-Garvey, Jul 22 2025

Status

approved