A007430 Inverse Moebius transform applied thrice to natural numbers. (Formerly M3750)

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

1,2

Comments

Equals row sums of triangle A140704. - Gary W. Adamson, May 24 2008

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

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).

Row sums of triangle A134676. - Gary W. Adamson, Nov 05 2007

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

Maple

with(numtheory); A007430:=proc(q) local a, b, c, j, k, n;
for n from 1 to q do a:=divisors(n); c:=0; for k from 1 to nops(a) do b:=divisors(a[k]);   c:=c+add(sigma(b[j]), j=1..nops(b)); od;  print(c); od; end: A007430(10^6); # Paolo P. Lava, May 07 2013

Mathematica

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

Prog

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

Crossrefs

Cf. A134676, A140704, A000005, A000203, A027750.

Sequence in context: A019070 A019071 A028285 * A118712 A130878 A104422

Adjacent sequences: A007427 A007428 A007429 * A007431 A007432 A007433

Keyword

nonn,easy,nice,mult

Author

N. J. A. Sloane

Status

approved