OFFSET
1,7
COMMENTS
Conjecture: (i) a(n) > 0 for all n > 5.
(ii) For n > 31, there is a positive integer k < n - 2 with phi(k) + phi(n-k)/2 a square. If n > 70 is not equal to 150, then phi(k) + phi(n-k) is a square for some 0 < k < n.
(iii) If n > 5, then phi(k) + phi(n-k)/2 is a triangular number for some 0 < k < n - 2. For each n = 20, 21, ... there is a positive integer k < n with (phi(k) + phi(n-k))/2 a triangular number.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
EXAMPLE
a(8) = 1 since 1 + phi(7)/2 = 1 + 3 = 2^2.
a(11) = 1 since 8 + phi(3)/2 = 8 + 1 = 3^2.
a(78) = 1 since 40 + phi(38)/2 = 40 + 9 = 7^2.
MATHEMATICA
SQ[n_]:=IntegerQ[Sqrt[n]]
p[n_, k_]:=SQ[k+EulerPhi[n-k]/2]
a[n_]:=Sum[If[p[n, k], 1, 0], {k, 1, n-3}]
Table[a[n], {n, 1, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 02 2014
STATUS
approved