A370998 2*a(n) = m is the least even number m such that all sums m + prime(k), k=1..n are composite.

1, 3, 11, 44, 44, 56, 56, 101, 101, 101, 359, 359, 359, 664, 664, 821, 821, 821, 866, 866, 866, 2623, 2623, 2623, 2623, 2944, 2944, 2944, 2944, 2944, 2944, 2944, 2944, 2944, 5171, 5171, 12839, 18833, 18833, 18833, 18833, 29947, 29947, 29947, 38002, 38002, 38002, 38002, 51551

Offset: 1

Author

Hugo Pfoertner, Mar 09 2024

Keywords

• nonn

Examples

			a(1) = 1: prime(1) = 2; 2 + 2*a(1) = 4 is the first composite.
a(2) = 3: m = 6; since all sums prime(1) + 2*x are even, any x can be chosen. prime(2) = 3, 3 + 6 = 9, whereas 3 + 1*2 and 3 + 2*2 are prime.
a(3) = 11: m = 22; for any even m < 22 at least one of 3 + m or 5 + m would be prime, e.g., 3+2=5, 3+4=7, 5+6=11, 3+8=11, 5+12=17, 5+14=19, 3+16=19, 5+18=23, 3+20=23, but 3+22=25 and 5+22 are composite.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..273

Crossrefs

Cf. A239392 (records).

Programs

Maple

R:= 1: P:= [2]: r:= 1:
for n from 2 to 100 do
  P:= [op(P), ithprime(n)];
  for k from r while ormap(isprime,P +~ 2*k) do od:
  R:= R, k; r:= k;
od:
R; # Robert Israel, Mar 09 2025

Python

from itertools import count
from sympy import prime, isprime
def A370998(n):
    ptuple = tuple(prime(k) for k in range(1,n+1))
    return next(filter(lambda m:not any(isprime(p+m) for p in ptuple),count(2,2)))>>1 # Chai Wah Wu, Mar 21 2024