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Totally Goldbach numbers: Positive integers n such that for all primes p < n-1 with p not dividing n, n-p is prime.
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%I #10 Oct 13 2017 15:48:53

%S 1,2,3,4,5,6,8,10,12,18,24,30

%N Totally Goldbach numbers: Positive integers n such that for all primes p < n-1 with p not dividing n, n-p is prime.

%C As Browers et al. point out, A141340 = A141341 union {7,14,16,36,42,48,60,90,210}, A020490 = A141341\{5} and A048597 = A141341\{5,10}. The authors show that the first strategy of Deshouillers et al. to establish a bound (of 10^520) for A141340 is sufficient for then determining the totally Goldbach numbers and "leads us naturally to interesting questions concerning primes in a fixed residue class".

%H J-M. Deshouillers, A. Granville, W. Narkiewicz and C. Pomerance, <a href="https://doi.org/10.1090/S0025-5718-1993-1202609-9">An upper bound in Goldbach's problem</a>, Math. Comp. 61 (1993), 209-213.

%H David van Golstein Brouwers, John Bamberg and Grant Cairns, <a href="http://www.austms.org.au/Publ/Gazette/2004/Sep04/brouwers.pdf">Totally Goldbach numbers and related conjectures</a>, The Australian Mathematical Society, Gazette, Volume 31 Number 4, September 2004.

%H <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>

%Y Cf. A020490, A048597, A141340.

%K fini,full,nonn

%O 1,2

%A _Rick L. Shepherd_, Jun 25 2008