A007432 Moebius transform applied thrice to natural numbers. (Formerly M0031)

1, -1, 0, 1, 2, 0, 4, 1, 3, -2, 8, 0, 10, -4, 0, 2, 14, -3, 16, 2, 0, -8, 20, 0, 13, -10, 8, 4, 26, 0, 28, 4, 0, -14, 8, 3, 34, -16, 0, 2, 38, 0, 40, 8, 6, -20, 44, 0, 31, -13, 0, 10, 50, -8, 16, 4, 0, -26, 56, 0, 58, -28, 12, 8, 20, 0, 64, 14

Offset

1,5

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n=1..1000

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / (108*n^2/Pi^6) for n = 1..1000000

N. J. A. Sloane, Transforms

Formula

Multiplicative with a(p^e) = Sum_{k=0..3} (-1)^k C(3,k)*p^(e-k)[e>=k];

Dirichlet g.f.: zeta(s-1)/zeta^3(s).

a(n) = Sum{d|n} tau_{-3}(d)*n/d = Sum{d|n} tau_{-2}(d)*phi(n/d), where tau_{-3} is A007428 and tau_{-2} is A007427. - Enrique Pérez Herrero, Jan 19 2013

Sum_{k=1..n} a(k) ~ 108 * n^2 / Pi^6. - Vaclav Kotesovec, Nov 04 2018

Mathematica

a[p_, e_] := Sum[ (-1)^k*Binomial[3, k]*p^(e - k), {k, 0, Min[e, 3]}]; a[n_] := Times @@ Apply[a, FactorInteger[n], {1}]; a[1] = 1; Table[ a[n], {n, 1, 68}] (* Jean-François Alcover, Dec 28 2011, after formula *)

Crossrefs

Cf. A007431.

Sequence in context: A347961 A020781 A327883 * A079124 A242071 A176910

Adjacent sequences: A007429 A007430 A007431 * A007433 A007434 A007435

Keyword

sign,easy,nice,mult

Author

N. J. A. Sloane

Status

approved