A334810 - OEIS

%I #18 Apr 24 2023 09:35:33

%S 0,0,0,0,1,0,1,0,1,4,1,3,2,1,1,4,7,2,4,3,1,4,3,4,6,5,2,2,0,1,8,7,8,3,

%T 8,5,5,7,5,6,7,2,8,4,3,1,7,14,10,7,4,6,3,7,8,11,14,8,6,5,3,8,14,10,7,

%U 6,11,13,15,10,7,7,9,10,11,8,9,10,7,9,13,9,13,10,9,6,6,8,7,3,2,9,10,10,10,10,8,15,9,20

%N The number of even numbers between 4 and 2p that cannot be written as the sum of two primes less than or equal to the n-th prime number p.

%C This sequence is related to the Goldbach conjecture: any even number (m) greater than 4 can be written as the sum of two odd primes (i.e., m = p1 + p2). For any given prime number p, if the restriction (p1, p2 <= p) is applied, some even numbers less than 2p may not be written as the sum of two prime numbers. The prime numbers corresponding to a(n)=0 in this sequence are the seven prime numbers listed in A301776.

%H Ya-Ping Lu, <a href="/A334810/b334810.txt">Table of n, a(n) for n = 1..100000</a>

%e a(1)=0. The 1st prime is 2. Even number 4 can be written as 2+2.

%e a(2)=0. The 2nd prime is 3. Even numbers 4 and 6 can be written as: 4=2+2 and 6=3+3.

%e a(3)=0. The 3rd prime is 5. Even numbers between 4 and 10 are: 4=2+2, 6=3+3, 8=3+5, and 10=5+5.

%e a(5)=1. The 5th prime is 11. Among the 10 even numbers between 4 and 22, only 20 cannot be written as the sum of two primes <= 11.

%e a(10)=4. The 10th prime is 29. Four even numbers (44, 50, 54, and 56) between 4 and 58 cannot be written as the sum of two primes <= 29.

%t a[n_] := Block[{p = Prime[n], r = Prime@ Range@ n}, Sum[Boole[{} == IntegerPartitions[2 k, {2}, r]], {k, 2, p}]]; Array[a, 83] (* _Giovanni Resta_, May 12 2020 *)

%Y Cf. A301776 (prime numbers p with the property that all even numbers n (2 < n <= 2p) are the sum of two primes <= p).

%K nonn

%O 1,10

%A _Ya-Ping Lu_, May 12 2020