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A007429 Inverse Moebius transform applied twice to natural numbers.
(Formerly M3249)
29
1, 4, 5, 11, 7, 20, 9, 26, 18, 28, 13, 55, 15, 36, 35, 57, 19, 72, 21, 77, 45, 52, 25, 130, 38, 60, 58, 99, 31, 140, 33, 120, 65, 76, 63, 198, 39, 84, 75, 182, 43, 180, 45, 143, 126, 100, 49, 285, 66, 152, 95, 165, 55, 232, 91, 234, 105, 124, 61, 385, 63, 132, 162, 247, 105, 260 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

REFERENCES

D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston, MA, 1976, p. 120.

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

LINKS

Travis Scholl, Table of n, a(n) for n = 1..100000 (terms 1 through 1000 were by T. D. Noe)

O. Bordelles, Mean values of generalized gcd-sum and lcm-sum functions, JIS 10 (2007) 07.9.2, series g_4 (with an apparently wrong D.g.f after equation 3).

N. J. A. Sloane, Transforms

FORMULA

a(n) = Sum_{d|n} sigma(d), Dirichlet convolution of A000203 and A000012. - Jason Earls (zevi_35711(AT)yahoo.com), Jul 07 2001

a(n) = Sum_{d|n} d*tau(n/d). - Vladeta Jovovic, Jul 31 2002

Multiplicative with a(p^e) = (p*(p^(e+1)-1)-(p-1)*(e+1))/(p-1)^2. - Vladeta Jovovic, Dec 25 2001

G.f.: Sum_{k>=1} sigma(k)*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003

Moebius transform of A007430. - Benoit Cloitre, Mar 03 2004

Dirichlet g.f.: zeta(s-1)*zeta^2(s).

Equals A051731^2 * [1, 2, 3,...]. Equals row sums of triangle A134577. - Gary W. Adamson, Nov 02 2007

Row sums of triangle A134699. - Gary W. Adamson, Nov 06 2007

a(n) = n * (Sum_{d|n} tau(d)/d) = n * (A276736(n) / A276737(n)). - Jaroslav Krizek, Sep 24 2016

MAPLE

A007429 := proc(n)

    add(numtheory[sigma](d), d=numtheory[divisors](n)) ;

end proc:

seq(A007429(n), n=1..100) ; # R. J. Mathar, Aug 28 2015

MATHEMATICA

f[n_] := Plus @@ DivisorSigma[1, Divisors@n]; Array[f, 52] (* Robert G. Wilson v, May 05 2010 *)

PROG

(PARI) j=[]; for(n=1, 200, j=concat(j, sumdiv(n, d, sigma(d)))); j

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

(PARI)

N=17; default(seriesprecision, N); x=z+O(z^(N+1))

c=sum(j=1, N, j*x^j);

t=1/prod(j=1, N, eta(x^(j))^(1/j))

t=log(t)

t=serconvol(t, c)

Vec(t)

/* Joerg Arndt, May 03 2008 */

(PARI) a(n)=sumdiv(n, d, sumdiv(d, t, t ) );  /* Joerg Arndt, Oct 07 2012 */

(Sage) def A(n): return sum(sigma(d) for d in n.divisors()) # Travis Scholl, Apr 14 2016

(MAGMA) [&+[SumOfDivisors(d): d in Divisors(n)]: n in [1..100]] // Jaroslav Krizek, Sep 24 2016

CROSSREFS

Cf. A134699.

Sequence in context: A126069 A147559 A206028 * A064945 A069820 A178823

Adjacent sequences:  A007426 A007427 A007428 * A007430 A007431 A007432

KEYWORD

nonn,easy,nice,mult

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified March 22 22:17 EDT 2017. Contains 283901 sequences.