A007429 Inverse Moebius transform applied twice to natural numbers.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice
• mult

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

a(n) is the sum of the sum-of-divisors of the divisors of n. - M. F. Hasler, Mar 29 2024

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.

Crossrefs

Cf. A000203, A007430, A134699, A152649, A280077.

Programs

Magma

[&+[SumOfDivisors(d): d in Divisors(n)]: n in [1..100]] // Jaroslav Krizek, Sep 24 2016

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 *)

PARI

A007429_upto(N)=vector(N,n, sumdiv(n,d, sigma(d))) \\ edited by M. F. Hasler, Mar 29 2024

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 */

Python

from math import prod
from sympy import factorint
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

Sage

def A(n): return sum(sigma(d) for d in n.divisors()) # Travis Scholl, Apr 14 2016

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