The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 60th year, we have over 367,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A007429 Inverse Moebius transform applied twice to natural numbers. (Formerly M3249) 59
 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 COMMENTS 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 REFERENCES 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). LINKS Travis Scholl, Table of n, a(n) for n = 1..100000 (terms 1 through 1000 were by T. D. Noe) Olivier 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. FORMULA 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 Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4/72 = 1.352904... (A152649). - Amiram Eldar, Oct 22 2022 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 *) 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 *) 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. A000203, A007430, A134699, A152649, A280077. Sequence in context: A147559 A322655 A206028 * A064945 A069820 A318446 Adjacent sequences: A007426 A007427 A007428 * A007430 A007431 A007432 KEYWORD nonn,easy,nice,mult AUTHOR N. J. A. Sloane STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 6 15:29 EST 2023. Contains 367610 sequences. (Running on oeis4.)