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.

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, 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, 1, 1

Offset

1,33

Comments

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.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..7000

Example

a(18) = 1 since 18 = 1 + 2 + 15 with phi(1) = 1^3, phi(2) = 1^3 and phi(15) = 2^3.
a(101) = 1 since 101 = 1 + 15 + 85 with phi(1) = 1^3, phi(15) = 2^3 and phi(85) = 4^3.
a(1613) = 1 since 1613 = 192 + 333 + 1088 with phi(192) = 4^3, phi(333) = 6^3 and phi(1088) = 8^3.

Mathematica

CQ[n_]:=IntegerQ[EulerPhi[n]^(1/3)]
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}]
Table[a[n], {n, 1, 70}]

Crossrefs

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

Sequence in context: A053694 A085862 A257392 * A024942 A210255 A283319

Adjacent sequences: A237120 A237121 A237122 * A237124 A237125 A237126

Keyword

nonn

Author

Zhi-Wei Sun, Feb 03 2014

Status

approved