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A233386 Number of ways to write n = i + j + k with 0 < i <= j <= k and i, j, k not all equal such that phi(i)*phi(j)*phi(k) is a cube. 7
0, 0, 0, 1, 1, 0, 0, 0, 1, 3, 2, 3, 4, 3, 4, 4, 4, 5, 3, 3, 1, 2, 6, 5, 5, 6, 7, 8, 8, 7, 8, 13, 7, 9, 4, 6, 8, 10, 7, 11, 14, 12, 8, 9, 10, 14, 12, 9, 9, 8, 8, 11, 8, 9, 19, 14, 12, 9, 11, 19, 12, 19, 10, 15, 13, 22, 18, 27, 22, 31, 20, 22, 18, 25, 25, 24, 18, 22, 19, 21, 24, 22, 30, 31, 35, 25, 28, 32, 23, 27, 28, 29, 23, 24, 30, 30, 29, 30, 33, 31 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,10

COMMENTS

Conjecture: For each k = 3, 4, ..., any integer n >= 3*k can be written as n_1 + n_2 + ... + n_k with n_1, n_2, ..., n_k positive and not all equal such that the product phi(n_1)*phi(n_2)*...*phi(n_k) is a k-th power.

We have verified this conjecture with k = 3 for n up to 10^5 and with k = 4, 5, 6 for n up to 30000.

See also A236998 for a similar conjecture with k = 2.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..1000

EXAMPLE

a(9) = 1 since 9 = 1 + 3 + 5 with phi(1)*phi(3)*phi(5) = 1*2*4 = 2^3.

a(21) = 1 since 21 = 5 + 8 + 8 with phi(5)*phi(8)*phi(8) = 4*4*4 = 4^3.

MATHEMATICA

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

p[i_, j_, k_]:=CQ[EulerPhi[i]*EulerPhi[j]*EulerPhi[k]]

a[n_]:=Sum[If[p[i, j, n-i-j], 1, 0], {i, 1, (n-1)/3}, {j, i, (n-i)/2}]

Table[a[n], {n, 1, 100}]

CROSSREFS

Cf. A000010, A000578, A236998.

Sequence in context: A322200 A028292 A256244 * A093407 A147658 A221529

Adjacent sequences:  A233383 A233384 A233385 * A233387 A233388 A233389

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Feb 02 2014

STATUS

approved

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Last modified March 31 19:37 EDT 2020. Contains 333151 sequences. (Running on oeis4.)