OFFSET
1,2
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
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
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
KEYWORD
nonn,easy,nice,mult
AUTHOR
STATUS
approved