A141340 -id:A141340 - cp's OEIS frontend

A141341 Totally Goldbach numbers: Positive integers k such that for all primes p < k-1 with p not dividing k, k-p is prime.

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

Offset: 1

Rick L. Shepherd, Jun 25 2008

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".

J-M. Deshouillers, A. Granville, W. Narkiewicz and C. Pomerance, An upper bound in Goldbach's problem, Math. Comp. 61 (1993), 209-213.
David van Golstein Brouwers, John Bamberg and Grant Cairns, Totally Goldbach numbers and related conjectures, The Australian Mathematical Society, Gazette, Volume 31 Number 4, September 2004.
Index entries for sequences related to Goldbach conjecture

Cf. A020490, A048597, A141340.

q[k_]:=AllTrue[k-Select[Prime[Range[PrimePi[k-2]]],!Divisible[k,#]&],PrimeQ];Select[Range[30],q[#]&] (* James C. McMahon, Jul 21 2025 *)

cp's OEIS Frontend

A287642 Duplicate of A141340.

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A141341 Totally Goldbach numbers: Positive integers k such that for all primes p < k-1 with p not dividing k, k-p is prime.

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