A216275 Fibonacci + Goldbach: a(1)=6, a(2)=8 and for n>=3, a(n)=g(a(n-1)) + g(a(n-2)), where for m>=3, g(2*m) is the maximal prime p < 2*m such that 2*m - p is prime.

6, 8, 8, 10, 12, 14, 18, 24, 32, 48, 72, 110, 174, 274, 438, 704, 1134, 1830, 2952, 4762, 7698, 12450, 20128, 32560, 52660, 85168, 137752, 222844, 360564, 583392, 943902, 1527222, 2471074, 3998274, 6469334, 10467566, 16936850, 27404300, 44341050, 71745324

Offset: 1

Author

Vladimir Shevelev, Mar 16 2013

Keywords

• nonn

Comments

Conjecture. lim a(n+1)/a(n)=phi as n goes to infinity (phi=golden ratio).

Examples

			Let n=6. Since a(4) = 10, a(5) = 12 and g(10) = g(12) = 7, then a(6) = 7 + 7 = 14.
		

Links

• Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000045, A002375, A025019, A216835.

Programs

Mathematica

a[1] = 6; a[2] = 8; g[n_] := Module[{tmp,k=1}, While[!PrimeQ[n-(tmp=NextPrime[n,-k])], k++]; tmp]; a[n_] := a[n] = g[a[n-1]] + g[a[n-2]]; Table[a[n], {n,1,100}]

Formula

For n>=5, a(n) = A216835(n-3) + A216835(n-4).