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A007429
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Inverse Moebius transform applied twice to natural numbers.
(Formerly M3249)
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57
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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, 55, 232, 91, 234, 105, 124, 61, 385, 63, 132, 162, 247, 105, 260
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OFFSET
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1,2
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COMMENTS
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Sum of the divisors d1 from the ordered pairs of divisors of n, (d1,d2) with d1<=d2, such that d1|d2. - Wesley Ivan Hurt, Mar 22 2022
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REFERENCES
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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
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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, Vol. 10 (2007), Article 07.9.2, series g_4 (with an apparently wrong D.g.f. after equation 3).
N. J. A. Sloane, Transforms.
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FORMULA
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a(n) = Sum_{d|n} sigma(d), Dirichlet convolution of A000203 and A000012. - Jason Earls, 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
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(sigma(k)/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 26 2018
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MAPLE
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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
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MATHEMATICA
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f[n_] := Plus @@ DivisorSigma[1, Divisors@n]; Array[f, 52] (* Robert G. Wilson v, May 05 2010 *)
f[p_, e_] := (p*(p^(e + 1) - 1) - (p - 1)*(e + 1))/(p - 1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 10 2022 *)
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PROG
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(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
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CROSSREFS
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Cf. A000203, A007430, A134699, A280077.
Sequence in context: A147559 A322655 A206028 * A064945 A069820 A318446
Adjacent sequences: A007426 A007427 A007428 * A007430 A007431 A007432
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KEYWORD
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nonn,easy,nice,mult
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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