

A000189


Number of solutions to x^3 == 0 (mod n).


14



1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 4, 5, 1, 9, 2, 1, 1, 1, 8, 1, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 2, 3, 1, 1, 4, 7, 5, 1, 2, 1, 9, 1, 4, 1, 1, 1, 2, 1, 1, 3, 16, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 5, 2, 1, 1, 1, 4, 9, 1, 1, 2, 1, 1, 1, 4, 1, 3
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OFFSET

1,4


COMMENTS

Shadow transform of the cubes A000578.  Michel Marcus, Jun 06 2013


LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000
H. Bottomley, Some Smarandachetype multiplicative sequences.
S. R. Finch and Pascal Sebah, Squares and Cubes Modulo n, arXiv:math/0604465 [math.NT], 20062016.
Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5(4) (1999), 138150. (ps, pdf); see Definition 7 for the shadow transform.
OEIS Wiki, Shadow transform.
N. J. A. Sloane, Transforms.


FORMULA

Multiplicative with a(p^e) = p^[2e/3].  David W. Wilson, Aug 01 2001
a(n) = n/A019555(n).  Petros Hadjicostas, Sep 15 2019


EXAMPLE

a(4) = 2 because 0^3 == 0, 1^3 == 1, 2^3 == 0, and 3^3 == 3 (mod 4); also, a(9) = 3 because 0^3 = 0, 3^3 == 0, and 6^3 = 0 (mod 9), while x^3 =/= 0 (mod 9) for x = 1, 2, 4, 5, 7, 8.  Petros Hadjicostas, Sep 16 2019


MATHEMATICA

Array[ Function[ n, Count[ Array[ PowerMod[ #, 3, n ]&, n, 0 ], 0 ] ], 100 ]


PROG

(PARI) a(n)=my(f=factor(n)); prod(i=1, #f[, 1], f[i, 1]^(2*f[i, 2]\3)) \\ Charles R Greathouse IV, Jun 06 2013


CROSSREFS

Cf. A000578, A019555.
Sequence in context: A220632 A125653 A104445 * A000190 A003557 A073752
Adjacent sequences: A000186 A000187 A000188 * A000190 A000191 A000192


KEYWORD

nonn,mult,easy


AUTHOR

N. J. A. Sloane


STATUS

approved



