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A237123 Number of ways to write n = i + j + k with 0 < i < j < k such that phi(i), phi(j) and phi(k) are all cubes, where phi(.) is Euler's totient function.

%I A237123 #14 Feb 04 2014 00:42:07
%S A237123 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,1,0,0,0,1,0,0,0,0,1,2,0,
%T A237123 0,1,2,1,0,1,2,1,0,0,1,2,2,1,0,0,2,1,0,0,2,1,0,0,1,1,1,0,0,0,1,1,0,0,
%U A237123 1,1
%N A237123 Number of ways to write n = i + j + k with 0 < i < j < k such that phi(i), phi(j) and phi(k) are all cubes, where phi(.) is Euler's totient function.
%C A237123 Conjecture: For each k = 2, 3, ... there is a positive integer s(k) such that any integer n >= s(k) can be written as i_1 + i_2 + ... + i_k with 0 < i_1 < i_2 < ... < i_k such that all those phi(i_1), phi(i_2), ..., phi(i_k) are k-th powers. In particular, we may take s(2) = 70640, s(3) = 935 and s(4) = 3273.
%H A237123 Zhi-Wei Sun, <a href="/A237123/b237123.txt">Table of n, a(n) for n = 1..7000</a>
%e A237123 a(18) = 1 since 18 = 1 + 2 + 15 with phi(1) = 1^3, phi(2) = 1^3 and phi(15) = 2^3.
%e A237123 a(101) = 1 since 101 = 1 + 15 + 85 with phi(1) = 1^3, phi(15) = 2^3 and phi(85) = 4^3.
%e A237123 a(1613) = 1 since 1613 = 192 + 333 + 1088 with phi(192) = 4^3, phi(333) = 6^3 and phi(1088) = 8^3.
%t A237123 CQ[n_]:=IntegerQ[EulerPhi[n]^(1/3)]
%t A237123 a[n_]:=Sum[If[CQ[i]&&CQ[j]&&CQ[n-i-j],1,0],{i,1,n/3-1},{j,i+1,(n-1-i)/2}]
%t A237123 Table[a[n],{n,1,70}]
%Y A237123 Cf. A000010, A000578, A039771, A233386, A236998.
%K A237123 nonn
%O A237123 1,33
%A A237123 _Zhi-Wei Sun_, Feb 03 2014