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Number of solutions to x^3 == 0 (mod n).
16

%I #58 Sep 09 2023 11:33:37

%S 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,

%T 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,

%U 1,1,1,12,1,1,5,2,1,1,1,4,9,1,1,2,1,1,1,4,1,3

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

%C Shadow transform of the cubes A000578. - _Michel Marcus_, Jun 06 2013

%H T. D. Noe, <a href="/A000189/b000189.txt">Table of n, a(n) for n = 1..10000</a>

%H Henry Bottomley, <a href="http://fs.gallup.unm.edu/Bottomley-Sm-Mult-Functions.htm">Some Smarandache-type multiplicative sequences</a>.

%H Steven R. Finch and Pascal Sebah, <a href="https://arxiv.org/abs/math/0604465">Squares and Cubes Modulo n</a>, arXiv:math/0604465 [math.NT], 2006-2016.

%H Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5(4) (1999), 138-150. (<a href="http://math.berkeley.edu/~halbeis/publications/psf/seq.ps">ps</a>, <a href="http://math.berkeley.edu/~halbeis/publications/pdf/seq.pdf">pdf</a>); see Definition 7 for the shadow transform.

%H Vaclav Kotesovec, <a href="/A000189/a000189.jpg">Graph - the asymptotic ratio (100000 terms)</a>.

%H OEIS Wiki, <a href="https://oeis.org/wiki/Shadow_transform">Shadow transform</a>.

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.

%F Multiplicative with a(p^e) = p^[2e/3]. - _David W. Wilson_, Aug 01 2001

%F a(n) = n/A019555(n). - _Petros Hadjicostas_, Sep 15 2019

%F Dirichlet g.f.: zeta(3*s-2) * Product_{p prime} (1 + 1/p^s + 1/p^(2*s-1)). - _Amiram Eldar_, Sep 09 2023

%F From _Vaclav Kotesovec_, Sep 09 2023: (Start)

%F Dirichlet g.f.: zeta(s) * zeta(2*s-1) * zeta(3*s-2) * Product_{p prime} (1 - 1/p^(2*s) - 1/p^(3*s-1) - 1/p^(4*s-2) + 1/p^(4*s-1) + 1/p^(5*s-2)).

%F Let f(s) = Product_{primes p} (1 - 1/p^(2*s) - 1/p^(3*s-1) - 1/p^(4*s-2) + 1/p^(4*s-1) + 1/p^(5*s-2)).

%F Sum_{k=1..n} a(k) ~ (f(1)*n/6) * (log(n)^2/2 + (6*gamma - 1 + f'(1)/f(1))*log(n) + 1 - 6*gamma + 11*gamma^2 - 14*sg1 + (6*gamma - 1)*f'(1)/f(1) + f''(1)/(2*f(1))), where

%F f(1) = Product_{primes p} (1 - 3/p^2 + 2/p^3) = A065473 = 0.2867474284344787341078927127898384464343318440970569956414778593366522431...,

%F f'(1) = f(1) * Sum_{primes p} 9*log(p) / (p^2 + p - 2) = f(1) * 4.1970213428422788650375569145777616746065054412058004220013841318980729375...,

%F f''(1) = f'(1)^2/f(1) + f(1) * Sum_{primes p} (-29*p^2 - 17*p + 1) * log(p)^2 / (p^2 + p - 2)^2 = f'(1)^2/f(1) + f(1) * (-21.3646716550082193262514333696570765444176783899223644201265894338042468...),

%F gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). (End)

%e 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

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

%t 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 *)

%o (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

%o (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

%Y Cf. A000578, A019555.

%K nonn,mult,easy

%O 1,4

%A _N. J. A. Sloane_