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 A007430 Inverse Moebius transform applied thrice to natural numbers. (Formerly M3750) 5

%I M3750

%S 1,5,6,16,8,30,10,42,24,40,14,96,16,50,48,99,20,120,22,128,60,70,26,

%T 252,46,80,82,160,32,240,34,219,84,100,80,384,40,110,96,336,44,300,46,

%U 224,192,130,50,594,76,230,120,256,56,410,112,420,132,160,62,768,64,170,240,466,128,420

%N Inverse Moebius transform applied thrice to natural numbers.

%C Equals row sums of triangle A140704. - _Gary W. Adamson_, May 24 2008

%C 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

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

%H T. D. Noe, <a href="/A007430/b007430.txt">Table of n, a(n) for n=1..10000</a>

%H O. Bordelles, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Bordelles2/bordelles61.html">Mean values of generalized gcd-sum and lcm-sum functions</a>, JIS 10 (2007) 07.9.2, sequence g_5.

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

%F a(n) = Sum_{d|n} sigma(d)*tau(n/d). - _Benoit Cloitre_, Mar 03 2004

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

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

%F Row sums of triangle A134676. - _Gary W. Adamson_, Nov 05 2007

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

%p with(numtheory); A007430:=proc(q) local a,b,c,j,k,n;

%p 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

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

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

%o (PARI) a(n)=if(n<1,0,direuler(p=2,n,1/(1-X)^3/(1-p*X))[n]) /* _Ralf Stephan_ */

%o (PARI) a(n)=sumdiv(n, x, sumdiv(x, y, sumdiv(y, z, z ) ) ); /* _Joerg Arndt_, Oct 07 2012 */

%o a007430 n = sum \$ zipWith (*) (map a000005 ds) (map a000203 \$ reverse ds)

%o where ds = a027750_row n

%o -- _Reinhard Zumkeller_, Aug 02 2014

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

%K nonn,easy,nice,mult

%O 1,2

%A _N. J. A. Sloane_

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Last modified April 18 22:08 EDT 2019. Contains 322237 sequences. (Running on oeis4.)