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Sum of sigma(j) for 1<=j<=10^n, where sigma(j) is the sum of the divisors of j.
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%I #53 Oct 23 2023 11:37:02

%S 1,87,8299,823081,82256014,8224740835,822468118437,82246711794796,

%T 8224670422194237,822467034112360628,82246703352400266400,

%U 8224670334323560419029,822467033425357340138978,82246703342420509396897774,8224670334241228180927002517

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

%H P. L. Patodia, Seth Troisi and Hiroaki Yamanouchi, <a href="/A072692/b072692.txt">Table of n, a(n) for n = 0..36</a> (terms a(0)-a(18) by P. L. Patodia and a(19)-a(24) by Seth Troisi)

%H Leonhard Euler, <a href="http://eulerarchive.maa.org/pages/E175.html">Découverte d'une loi tout extraordinaire des nombres par rapport à la somme de leurs diviseurs</a>, 1747, The Euler Archive, (Eneström Index) E175.

%H P. L. Patodia (pannalal(AT)usa.net), <a href="/A072692/a072692.txt">PARI program for A072692 and A024916</a>

%F 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.

%e 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)).

%o (PARI) for(m=0,10,print1(sum(n=1,k=10^m,n*(k\n)),",")) \\ Improved by _M. F. Hasler_, Apr 18 2015

%o (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

%o (Python)

%o from math import isqrt

%o 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

%o (PARI) A072692(n)=A024916(10^n) \\ This is very efficient, using efficient code of A024916. - _M. F. Hasler_, Apr 18 2015

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

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

%K nonn

%O 0,2

%A _Rick L. Shepherd_, Jul 02 2002

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

%E Corrected by _N. J. A. Sloane_, Jun 08 2008, following suggestions from _Don Reble_ and _David W. Wilson_