A007430 Inverse Moebius transform applied thrice to natural numbers.

1, 5, 6, 16, 8, 30, 10, 42, 24, 40, 14, 96, 16, 50, 48, 99, 20, 120, 22, 128, 60, 70, 26, 252, 46, 80, 82, 160, 32, 240, 34, 219, 84, 100, 80, 384, 40, 110, 96, 336, 44, 300, 46, 224, 192, 130, 50, 594, 76, 230, 120, 256, 56, 410, 112, 420, 132, 160, 62, 768, 64, 170, 240, 466, 128, 420

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice
• mult

Comments

a(n) = A000027(n) * A000012(n) * A000012(n) * A000012(n) = A000027(n) * A000012(n) * A000005(n) = A000203(n) * A000005(n) = A000203(n) * A000012(n) * A000012(n) = A007429(n) * A000012(n), where operation * denotes Dirichlet convolution for n >= 1. Dirichlet convolution of functions b(n), c(n) is function a(n) = b(n) * c(n) = Sum_{d|n} b(d)*c(n/d). - Jaroslav Krizek, Mar 20 2009

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..10000
• O. Bordelles, Mean values of generalized gcd-sum and lcm-sum functions, JIS 10 (2007) 07.9.2, sequence g_5.
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000005, A000203, A027750.

Programs

Haskell

a007430 n = sum $ zipWith (*) (map a000005 ds) (map a000203 $ reverse ds)
            where ds = a027750_row n
-- Reinhard Zumkeller, Aug 02 2014

Mathematica

a[n_] := Total[ DivisorSigma[1, #]*DivisorSigma[0, n/#]& /@ Divisors[n]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Nov 15 2011 *)

PARI

a(n)=sumdiv(n,d,sigma(d)*numdiv(n/d))

PARI

a(n)=if(n<1,0,direuler(p=2,n,1/(1-X)^3/(1-p*X))[n]) /* Ralf Stephan */

PARI

a(n)=sumdiv(n, x, sumdiv(x, y, sumdiv(y, z, z ) ) ); /* Joerg Arndt, Oct 07 2012 */

Formula

a(n) = Sum_{d|n} sigma(d)*tau(n/d). - Benoit Cloitre, Mar 03 2004

Multiplicative with a(p^e) = Sum_{k=0..e} binomial(e-k+2, e-k)*p^k.

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

Sum_{k=1..n} a(k) ~ Pi^6 * n^2 / 432. - Vaclav Kotesovec, Nov 06 2018