A370998 - OEIS

%I #12 Mar 21 2024 14:24:49

%S 1,3,11,44,44,56,56,101,101,101,359,359,359,664,664,821,821,821,866,

%T 866,866,2623,2623,2623,2623,2944,2944,2944,2944,2944,2944,2944,2944,

%U 2944,5171,5171,12839,18833,18833,18833,18833,29947,29947,29947,38002,38002,38002,38002,51551

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

%e a(1) = 1: prime(1) = 2; 2 + 2*a(1) = 4 is the first composite.

%e 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.

%e 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.

%o (Python)

%o from itertools import count

%o from sympy import prime, isprime

%o def A370998(n):

%o     ptuple = tuple(prime(k) for k in range(1,n+1))

%o     return next(filter(lambda m:not any(isprime(p+m) for p in ptuple),count(2,2)))>>1 # _Chai Wah Wu_, Mar 21 2024

%Y Cf. A239392 (records).

%K nonn

%O 1,2

%A _Hugo Pfoertner_, Mar 09 2024