OFFSET
1,3
COMMENTS
Conjecture: (i) a(n) > 0 for all n > 1. Also, any n > 12 can be written as k + m (k > 0 and m > 2) with 6*k - 1, 6*k + 1 and k + phi(m)/2 all prime.
(ii) Each integer n > 34 can be written as p + q (q > 0) with p and p + phi(q) both prime. Also, any integer n > 14 can be written as p + q (q > 2) with p, p + 6 and p + phi(q)/2 all prime.
Clearly, part (i) of the conjecture implies that any integer n > 1 can be written as p + m - phi(m), where p is a prime and m is a positive integer.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
EXAMPLE
a(17) = 1 since 17 = 7 + 10 with 6*7 - 1 = 41, 6*7 + 1 = 43 and 7 + phi(10) = 7 + 4 = 11 all prime.
a(486) = 1 since 486 = 325 + 161 with 6*325 - 1 = 1949, 6*325 + 1 = 1951 and 325 + phi(161) = 325 + 132 = 457 all prime.
MATHEMATICA
p[n_, k_]:=PrimeQ[6k-1]&&PrimeQ[6k+1]&&PrimeQ[k+EulerPhi[n-k]]
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, Feb 02 2014
STATUS
approved