A237123 - OEIS

%I #14 Feb 04 2014 00:42:07

%S 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 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 1,1

%N 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 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 Zhi-Wei Sun, <a href="/A237123/b237123.txt">Table of n, a(n) for n = 1..7000</a>

%e a(18) = 1 since 18 = 1 + 2 + 15 with phi(1) = 1^3, phi(2) = 1^3 and phi(15) = 2^3.

%e a(101) = 1 since 101 = 1 + 15 + 85 with phi(1) = 1^3, phi(15) = 2^3 and phi(85) = 4^3.

%e a(1613) = 1 since 1613 = 192 + 333 + 1088 with phi(192) = 4^3, phi(333) = 6^3 and phi(1088) = 8^3.

%t CQ[n_]:=IntegerQ[EulerPhi[n]^(1/3)]

%t 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 Table[a[n],{n,1,70}]

%Y Cf. A000010, A000578, A039771, A233386, A236998.

%K nonn

%O 1,33

%A _Zhi-Wei Sun_, Feb 03 2014