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A007429 Inverse Moebius transform applied twice to natural numbers.

%I A007429 M3249 #101 Mar 30 2024 02:35:50
%S A007429 1,4,5,11,7,20,9,26,18,28,13,55,15,36,35,57,19,72,21,77,45,52,25,130,
%T A007429 38,60,58,99,31,140,33,120,65,76,63,198,39,84,75,182,43,180,45,143,
%U A007429 126,100,49,285,66,152,95,165,55,232,91,234,105,124,61,385,63,132,162,247,105,260
%N A007429 Inverse Moebius transform applied twice to natural numbers.
%C A007429 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
%C A007429 a(n) is the sum of the sum-of-divisors of the divisors of n. - _M. F. Hasler_, Mar 29 2024
%D A007429 David M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston, MA, 1976, p. 120.
%D A007429 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007429 Travis Scholl, <a href="/A007429/b007429.txt">Table of n, a(n) for n = 1..100000</a> (terms 1 through 1000 were by T. D. Noe)
%H A007429 Olivier Bordelles, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Bordelles2/bordelles61.html">Mean values of generalized gcd-sum and lcm-sum functions</a>, JIS, Vol. 10 (2007), Article 07.9.2, series g_4 (with an apparently wrong D.g.f. after equation 3).
%H A007429 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.
%F A007429 a(n) = Sum_{d|n} sigma(d), Dirichlet convolution of A000203 and A000012. - _Jason Earls_, Jul 07 2001
%F A007429 a(n) = Sum_{d|n} d*tau(n/d). - _Vladeta Jovovic_, Jul 31 2002
%F A007429 Multiplicative with a(p^e) = (p*(p^(e+1)-1)-(p-1)*(e+1))/(p-1)^2. - _Vladeta Jovovic_, Dec 25 2001
%F A007429 G.f.: Sum_{k>=1} sigma(k)*x^k/(1-x^k). - _Benoit Cloitre_, Apr 21 2003
%F A007429 Moebius transform of A007430. - _Benoit Cloitre_, Mar 03 2004
%F A007429 Dirichlet g.f.: zeta(s-1)*zeta^2(s).
%F A007429 Equals A051731^2 * [1, 2, 3, ...]. Equals row sums of triangle A134577. - _Gary W. Adamson_, Nov 02 2007
%F A007429 Row sums of triangle A134699. - _Gary W. Adamson_, Nov 06 2007
%F A007429 a(n) = n * (Sum_{d|n} tau(d)/d) = n * (A276736(n) / A276737(n)). - _Jaroslav Krizek_, Sep 24 2016
%F A007429 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
%F A007429 Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4/72 = 1.352904... (A152649). - _Amiram Eldar_, Oct 22 2022
%p A007429 A007429 := proc(n)
%p A007429     add(numtheory[sigma](d),d=numtheory[divisors](n)) ;
%p A007429 end proc:
%p A007429 seq(A007429(n),n=1..100) ; # _R. J. Mathar_, Aug 28 2015
%t A007429 f[n_] := Plus @@ DivisorSigma[1, Divisors@n]; Array[f, 52] (* _Robert G. Wilson v_, May 05 2010 *)
%t A007429 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 *)
%o A007429 (PARI) A007429_upto(N)=vector(N,n, sumdiv(n,d, sigma(d))) \\ edited by _M. F. Hasler_, Mar 29 2024
%o A007429 (PARI) a(n)=if(n<1,0,direuler(p=2,n,1/(1-X)^2/(1-p*X))[n]) \\ _Ralf Stephan_
%o A007429 (PARI)
%o A007429 N=17; default(seriesprecision,N); x=z+O(z^(N+1))
%o A007429 c=sum(j=1,N,j*x^j);
%o A007429 t=1/prod(j=1,N, eta(x^(j))^(1/j))
%o A007429 t=log(t)
%o A007429 t=serconvol(t,c)
%o A007429 Vec(t)
%o A007429 /* _Joerg Arndt_, May 03 2008 */
%o A007429 (PARI) a(n)=sumdiv(n,d, sumdiv(d,t, t ) );  /* _Joerg Arndt_, Oct 07 2012 */
%o A007429 (Sage) def A(n): return sum(sigma(d) for d in n.divisors()) # _Travis Scholl_, Apr 14 2016
%o A007429 (Magma) [&+[SumOfDivisors(d): d in Divisors(n)]: n in [1..100]] // _Jaroslav Krizek_, Sep 24 2016
%o A007429 (Python)
%o A007429 from math import prod
%o A007429 from sympy import factorint
%o A007429 def A007429(n): return prod((p*(p**(e+1)-1)-(p-1)*(e+1))//(p-1)**2 for p,e in factorint(n).items()) # _Chai Wah Wu_, Mar 28 2024
%Y A007429 Cf. A000203, A007430, A134699, A152649, A280077.
%K A007429 nonn,easy,nice,mult
%O A007429 1,2
%A A007429 _N. J. A. Sloane_