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A007429
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Inverse Moebius transform applied twice to natural numbers.
(Formerly M3249)
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59
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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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David 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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FORMULA
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Multiplicative with a(p^e) = (p*(p^(e+1)-1)-(p-1)*(e+1))/(p-1)^2. - Vladeta Jovovic, Dec 25 2001
Dirichlet g.f.: zeta(s-1)*zeta^2(s).
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
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4/72 = 1.352904... (A152649). - Amiram Eldar, Oct 22 2022
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MAPLE
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add(numtheory[sigma](d), d=numtheory[divisors](n)) ;
end proc:
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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)
(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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KEYWORD
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nonn,easy,nice,mult
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AUTHOR
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STATUS
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approved
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