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A066766 Decimal expansion of Sum_{k>=1} sigma(k)/2^k where sigma(k) is the sum of divisors of k, 1 <= d <= k.

%I A066766 #37 Oct 16 2022 07:59:46
%S A066766 2,7,4,4,0,3,3,8,8,8,7,5,9,4,8,8,3,6,0,4,8,0,2,1,4,8,9,1,4,9,2,2,7,2,
%T A066766 1,6,4,3,1,1,4,2,8,9,8,1,3,1,9,6,3,9,3,1,7,8,4,8,5,2,8,8,8,4,7,3,7,9,
%U A066766 1,2,2,8,3,2,6,3,8,9,5,6,8,8,5,6,6,2,5,2,3,1,0,7,1,2,5,0,6,8,8,7,7,3,7,4,0
%N A066766 Decimal expansion of Sum_{k>=1} sigma(k)/2^k where sigma(k) is the sum of divisors of k, 1 <= d <= k.
%D A066766 Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 354-361.
%H A066766 Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/dig/dig.html">Digital Search Tree Constants</a> [Broken link]
%H A066766 Steven R. Finch, <a href="http://web.archive.org/web/20010413214937/http://www.mathsoft.com:80/asolve/constant/dig/dig.html">Digital Search Tree Constants</a> [From the Wayback machine]
%F A066766 Equals Sum_{k>=1} k/(2^k - 1). - _Amiram Eldar_, Jun 22 2020
%F A066766 Faster converging series: Sum_{n >= 1} (1/2)^(n^2) * (n*(4^n - 1) + 2^n)/(2^n - 1)^2. - _Peter Bala_, Jan 19 2021
%F A066766 From _Amiram Eldar_, Oct 16 2022: (Start)
%F A066766 Equals Sum_{k>=1} 2^k/(2^k - 1)^2.
%F A066766 Equals A065442 + A065443. (End)
%e A066766 2.74403388875948836048021489149227216431142898131963931784...
%p A066766 evalf( add( (1/2)^(n^2) * (n*(4^n - 1) + 2^n)/(2^n - 1)^2, n = 1..20), 100); # _Peter Bala_, Jan 19 2021
%t A066766 RealDigits[Sum[n/(2^n - 1), {n, 1, 500}], 10, 100][[1]] (* _Amiram Eldar_, Jun 22 2020 *)
%o A066766 (PARI) smv(v)= s=0; for(i=1,matsize(v)[2],s=s+v[i]); s
%o A066766 A066766(n)= sm=0; for(j=1,n,sm=sm+smv(divisors(j)/2^j)); sm*1.0
%o A066766 (PARI) suminf(k=1, sigma(k)/2^k) \\ _Michel Marcus_, Apr 27 2018
%Y A066766 Cf. A000203, A065442, A065443, A066772, A335763.
%K A066766 nonn,cons
%O A066766 1,1
%A A066766 _Randall L Rathbun_, Jan 16 2002
%E A066766 Name corrected by _Paul D. Hanna_, Apr 26 2018