OFFSET
1,4
COMMENTS
Conjecture: a(n) > 0 except for n = 1, 8.
Clearly, this implies Goldbach's conjecture.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
EXAMPLE
a(3) = 1 since 2 + phi(1) = 3 and 2*3 - 3 = 3 are both prime.
a(20) = 1 since 11 + phi(9) = 17 and 2*20 - 17 = 23 are both prime.
a(22) = 1 since 1 + phi(21) = 13 and 2*22 - 13 = 31 are both prime.
a(24) = 1 since 9 + phi(15) = 17 and 2*24 - 17 = 31 are both prime.
a(76) = 1 since 67 + phi(9) = 73 and 2*76 - 73 = 79 are both prime.
MATHEMATICA
f[n_, k_]:=k+EulerPhi[n-k]
p[n_, k_]:=PrimeQ[f[n, k]]&&PrimeQ[2n-f[n, k]]
a[n_]:=a[n]=Sum[If[p[n, k], 1, 0], {k, 1, n-1}]
Table[a[n], {n, 1, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Dec 30 2013
STATUS
approved