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A358798 a(1) = 2, a(2) = 3; for n > 2, a(n) is the smallest prime that can be appended to the sequence so that the smallest even number >= 4 that cannot be generated as the sum of two (not necessarily distinct) terms from {a(1), ..., a(n-1)} can be generated from {a(1), ..., a(n)}. 1

%I #31 Feb 17 2023 21:55:57

%S 2,3,5,7,11,13,17,19,23,31,29,37,41,47,43,61,53,67,59,73,71,83,89,79,

%T 97,101,103,109,107,113,127,131,139,137,149,151,163,157,179,167,173,

%U 181,193,191,211,197,199,227,233,223,229,239,251,241,277,257,271,263

%N a(1) = 2, a(2) = 3; for n > 2, a(n) is the smallest prime that can be appended to the sequence so that the smallest even number >= 4 that cannot be generated as the sum of two (not necessarily distinct) terms from {a(1), ..., a(n-1)} can be generated from {a(1), ..., a(n)}.

%C Related to the Goldbach conjecture. Proving that this sequence is non-terminating would prove the Goldbach conjecture.

%H Travis J Weber, <a href="/A358798/b358798.txt">Table of n, a(n) for n = 1..10002</a>

%H Travis J Weber, <a href="https://github.com/thetravisweber/goldbach/blob/main/Consecutive%20Prime%20list.ipynb">Python notebook code</a>

%e For n=5, the sequence a(1..4) is 2, 3, 5, 7. The even numbers 4..14 are sums of two terms and 16 is the smallest even number not a sum. The smallest prime allowing 16 to be made is 11, by 5 + 11 = 16. Thus, a(5) = 11.

%e For n=106, the smallest even number that is not the sum of two terms a(1..105) is 1082. The smallest prime allowing 1082 to be made is 541, by 541 + 541 = 1082. Thus, a(106) = 541.

%K nonn

%O 1,1

%A _Travis J Weber_, Dec 05 2022

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Last modified February 26 05:55 EST 2024. Contains 370335 sequences. (Running on oeis4.)