A237524 Number of ordered ways to write n = i + j + k with 0 < i <= j <= k such that phi(i*j*k) is a cube, where phi(.) is Euler's totient function.

0, 0, 1, 1, 0, 0, 0, 1, 4, 3, 2, 2, 1, 1, 1, 1, 2, 5, 2, 3, 2, 3, 6, 5, 4, 4, 4, 5, 4, 5, 4, 6, 6, 5, 5, 9, 6, 10, 8, 7, 7, 5, 5, 4, 11, 10, 8, 10, 5, 8, 8, 10, 10, 8, 11, 16, 11, 13, 14, 16, 18, 19, 18, 16, 24, 19, 21, 18, 15, 21, 9, 15, 14, 13, 15, 18, 19, 20, 15, 19

Offset: 1

Author

Zhi-Wei Sun, Feb 09 2014

Keywords

• nonn

Comments

Conjecture: For each k = 3, 4, ..., any integer n > 2*k + 1 can be written as a sum of k positive integers n_1, n_2, ..., n_k such that phi(n_1*n_2*...*n_k) is a k-th power.

Note that 2*k + 2 = (k-1)*2 + 4 with phi(2^(k-1)*4) = 2^k.

See also A237523 for a similar conjecture with k = 2.

Examples

			a(4) = 1 since 4 = 1 + 1 + 2 with phi(1*1*2) = 1^3.
a(13) = 1 since 13 = 1 + 2 + 10 with phi(1*2*10) = 2^3.
a(16) = 1 since 16 = 4 + 4 + 8 with phi(4*4*8) = phi(2^7) = 4^3.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..1500

Crossrefs

Cf. A000010, A000578, A233386, A237050, A237123, A237523.

Programs

Mathematica

CQ[n_]:=IntegerQ[n^(1/3)]
q[n_]:=CQ[EulerPhi[n]]
a[n_]:=Sum[If[q[i*j(n-i-j)],1,0],{i,1,n/3},{j,i,(n-i)/2}]
Table[a[n],{n,1,80}]