OFFSET
1,10
COMMENTS
Conjecture: For each k = 3, 4, ..., any integer n >= 3*k can be written as n_1 + n_2 + ... + n_k with n_1, n_2, ..., n_k positive and not all equal such that the product phi(n_1)*phi(n_2)*...*phi(n_k) is a k-th power.
We have verified this conjecture with k = 3 for n up to 10^5 and with k = 4, 5, 6 for n up to 30000.
See also A236998 for a similar conjecture with k = 2.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..1000
EXAMPLE
a(9) = 1 since 9 = 1 + 3 + 5 with phi(1)*phi(3)*phi(5) = 1*2*4 = 2^3.
a(21) = 1 since 21 = 5 + 8 + 8 with phi(5)*phi(8)*phi(8) = 4*4*4 = 4^3.
MATHEMATICA
CQ[n_]:=IntegerQ[n^(1/3)]
p[i_, j_, k_]:=CQ[EulerPhi[i]*EulerPhi[j]*EulerPhi[k]]
a[n_]:=Sum[If[p[i, j, n-i-j], 1, 0], {i, 1, (n-1)/3}, {j, i, (n-i)/2}]
Table[a[n], {n, 1, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 02 2014
STATUS
approved