A072692 - OEIS

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

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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. SeeA072691for Pi^2/12. Observe thatA025281also 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 byM. 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

defA072692(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 ofA024916.-M. F. Hasler,Apr 18 2015

CROSSREFS

Compare withA049000.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 byN. J. A. Sloane,Jun 08 2008, following suggestions fromDon RebleandDavid W. Wilson

STATUS

approved