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

Author

Rick L. Shepherd, Jun 25 2008

Keywords

• fini
• full
• nonn

Comments

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".

Links

• 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

Crossrefs

Cf. A020490, A048597, A141340.

Programs

Mathematica

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