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A308040 Numbers k such that k - prevprime(k-1) is prime where prevprime(n) is the largest prime < n.

%I A308040 #6 Feb 16 2025 08:33:55
%S A308040 4,5,6,7,8,9,10,12,13,14,15,16,18,19,20,21,22,24,25,26,28,30,31,32,33,
%T A308040 34,36,38,39,40,42,43,44,45,46,48,49,50,52,54,55,56,58,60,61,62,63,64,
%U A308040 66,68,69,70,72,73,74,75,76,78,80,81,82,84,85,86,88,90,91,92,94,96,99,100
%N A308040 Numbers k such that k - prevprime(k-1) is prime where prevprime(n) is the largest prime < n.
%C A308040 Contains all odd numbers k >= 5 such that k - 2 is prime.
%C A308040 If Goldbach's conjecture is true, the sequence contains an even number k iff there exists a Goldbach partition of k that includes the largest prime < k - 1. This sequence agrees with A014091 (numbers that are the sum of two primes) up to k = 96, but does not include 98 since the largest prime strictly below 97 is 89, which is paired with 9 (and thus, not a Goldbach partition).
%H A308040 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldbachPartition.html">Goldbach Partition</a>
%H A308040 Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach%27s_conjecture">Goldbach's conjecture</a>
%H A308040 <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%H A308040 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%t A308040 Select[Range[4, 100], PrimeQ[# - NextPrime[# - 1, -1]] &]
%Y A308040 Cf. A014091.
%K A308040 nonn
%O A308040 1,1
%A A308040 _Wesley Ivan Hurt_, May 10 2019