A072692 Sum of sigma(j) for 1<=j<=10^n, where sigma(j) is the sum of the divisors of j.

1, 87, 8299, 823081, 82256014, 8224740835, 822468118437, 82246711794796, 8224670422194237, 822467034112360628, 82246703352400266400, 8224670334323560419029, 822467033425357340138978, 82246703342420509396897774, 8224670334241228180927002517

Offset

0,2

Links

P. L. Patodia, Seth Troisi and Hiroaki Yamanouchi, Table of n, a(n) for n = 0..36 (terms a(0)-a(18) by P. L. Patodia and a(19)-a(24) by Seth Troisi)

Leonhard Euler, Découverte d'une loi tout extraordinaire des nombres par rapport à la somme de leurs diviseurs, 1747, The Euler Archive, (Eneström Index) E175.

P. L. Patodia (pannalal(AT)usa.net), PARI program for A072692 and A024916

Formula

Asymptotic formula: a(n) ~ Pi^2/12 * 10^2n. See A072691 for Pi^2/12. Observe that A025281 also contains that constant in its asymptotic formula.

Example

For n=1, the sum of sigma(j) for j<=10 is 1+3+4+7+6+12+8+15+13+18=87, so a(1)=87 (=69+18=A049000(1)+A046915(1)).

Prog

(PARI) for(m=0, 10, print1(sum(n=1, k=10^m, n*(k\n)), ", ")) \\ Improved by M. F. Hasler, Apr 18 2015
(Python) [(i, sum([d*(10**i//d) for d in range(1, 10**i+1)])) for i in range(8)] # Seth A. Troisi, Jun 27 2010
(Python)
from math import isqrt
def A072692(n): return -(s:=isqrt(m:=10**n))**2*(s+1)+sum((q:=m//k)*((k<<1)+q+1) for k in range(1, s+1))>>1 # Chai Wah Wu, Oct 23 2023
(PARI) A072692(n)=A024916(10^n) \\ This is very efficient, using efficient code of A024916. - M. F. Hasler, Apr 18 2015

Crossrefs

Compare with A049000. Note that a(n) = A049000(n) + A046915(n).

Cf. A000203 (sigma(n)), A072691 (Pi^2/12), A049000, A046915, A024916, A025281.

Sequence in context: A017750 A291130 A183040 * A287590 A133391 A298832

Adjacent sequences: A072689 A072690 A072691 * A072693 A072694 A072695

Keyword

nonn

Author

Rick L. Shepherd, Jul 02 2002

Extensions

More terms from P L Patodia (pannalal(AT)usa.net), Jan 11 2008, Jun 25 2008

Corrected by N. J. A. Sloane, Jun 08 2008, following suggestions from Don Reble and David W. Wilson

Status

approved