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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified July 7 12:08 EDT 2022. Contains 355148 sequences. (Running on oeis4.)