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A000189
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Number of solutions to x^3 == 0 (mod n).
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15
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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
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1,4
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COMMENTS
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Shadow transform of the cubes A000578. - Michel Marcus, Jun 06 2013
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..10000
Henry Bottomley, Some Smarandache-type multiplicative sequences.
Steven R. Finch and Pascal Sebah, Squares and Cubes Modulo n, arXiv:math/0604465 [math.NT], 2006-2016.
Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5(4) (1999), 138-150. (ps, pdf); see Definition 7 for the shadow transform.
OEIS Wiki, Shadow transform.
N. J. A. Sloane, Transforms.
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FORMULA
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Multiplicative with a(p^e) = p^[2e/3]. - David W. Wilson, Aug 01 2001
a(n) = n/A019555(n). - Petros Hadjicostas, Sep 15 2019
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EXAMPLE
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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
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MATHEMATICA
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Array[ Function[ n, Count[ Array[ PowerMod[ #, 3, n ]&, n, 0 ], 0 ] ], 100 ]
f[p_, e_] := p^Floor[2*e/3]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2020 *)
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PROG
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(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
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + X + p*X^2)/(1 - p^2*X^3))[n], ", ")) \\ Vaclav Kotesovec, Aug 30 2021
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CROSSREFS
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Cf. A000578, A019555.
Sequence in context: A220632 A125653 A104445 * A000190 A348037 A003557
Adjacent sequences: A000186 A000187 A000188 * A000190 A000191 A000192
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KEYWORD
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nonn,mult,easy
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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