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

1,9

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.

Links

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

Example

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.

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}]

Crossrefs

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

Sequence in context: A108438 A347738 A082504 * A266143 A171623 A117462

Adjacent sequences: A237521 A237522 A237523 * A237525 A237526 A237527

Keyword

nonn

Author

Zhi-Wei Sun, Feb 09 2014

Status

approved