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A007429
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Inverse Moebius transform applied twice to natural numbers.
(Formerly M3249)
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18
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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).
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..1000
N. J. A. Sloane, Transforms
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FORMULA
| a(n) = Sum_{d|n} sigma(d). - Jason Earls (zevi_35711(AT)yahoo.com), Jul 07 2001
a(n) = Sum_{d|n} d*tau(n/d). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 31 2002
Multiplicative with a(p^e) = (p*(p^(e+1)-1)-(p-1)*(e+1))/(p-1)^2. - Vladeta Jovovic (vladeta(AT)eunet.rs), Dec 25 2001
G.f.: sum(k>=1, sigma(k)*x^k/(1-x^k)). - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 21 2003
Moebius transform of A007430. - Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 03 2004
Dirichlet g.f.: zeta(x-1)zeta^2(x)
Equals A051731^2 * [1, 2, 3,...]. Equals row sums of triangle A134577. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 02 2007
Row sums of triangle A134699 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 06 2007
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MATHEMATICA
| f[n_] := Plus @@ DivisorSigma[1, Divisors@n]; Array[f, 52] [From Robert G. Wilson v (rgwv(AT)rgwv.com), May 05 2010]
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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]) (from R. Stephan)
(PARI from Joerg Arndt (arndt(AT)jjj.de), May 03, 2008)
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)
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CROSSREFS
| Cf. A134699.
Sequence in context: A196270 A126069 A147559 * A206028 A064945 A069820
Adjacent sequences: A007426 A007427 A007428 * A007430 A007431 A007432
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KEYWORD
| nonn,easy,nice,mult
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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