OFFSET
1,7
COMMENTS
Conjecture: (i) a(n) > 0 for all n > 8.
(ii) If n > 20, then phi(k)*phi(n-k) + 1 is a square for some 0 < k < n/2.
(iii) If n > 1 is not among 4, 7, 60, 199, 267, then k*phi(n-k) is a square for some 0 < k < n.
We have verified part (i) of the conjecture for n up to 2*10^6.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
EXAMPLE
a(17) = 1 since phi(5)*phi(12) = 4*4 = 4^2.
a(24) = 1 since phi(4)*phi(20) = 2*8 = 4^2.
a(56) = 1 since phi(8)*phi(48) = 4*16 = 8^2.
MATHEMATICA
SQ[n_]:=IntegerQ[Sqrt[n]]
p[n_, k_]:=SQ[EulerPhi[k]*EulerPhi[n-k]]
a[n_]:=Sum[If[p[n, k], 1, 0], {k, 1, (n-1)/2}]
Table[a[n], {n, 1, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 02 2014
STATUS
approved