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Decimal expansion of Sum_{k >= 1} 1/k^sigma_*(k) where sigma_*(n) is the sum of the anti-divisors of n.
2

%I #26 Oct 25 2015 17:39:25

%S 2,1,2,7,8,2,7,8,0,2,4,2,5,0,7,1,7,8,3,0,4,4,1,3,1,7,4,6,9,6,6,0,9,9,

%T 2,6,2,4,5,0,7,7,3,5,3,0,8,3,4,1,9,8,9,7,3,0,9,4,3,0,6,8,3,7,1,7,1,8,

%U 7,1,8,2,8,4,3,0,3,2,7,1,4,2,5,6,4,8

%N Decimal expansion of Sum_{k >= 1} 1/k^sigma_*(k) where sigma_*(n) is the sum of the anti-divisors of n.

%C Continued fraction (2,7,1,4,1,1,1,6,4,1,11,1,2...).

%e 1/1^sigma*(1)+ 1/2^sigma*(2) + 1/3^sigma*(3) + 1/4^sigma*(4) + 1/5^sigma*(5) + 1/6^sigma*(6) + ... = 1/1^0 + 1/2^0 + 1/3^2 + 1/4^3 + 1/5^5 + 1/6^4 + ... = 2.12782780242507..

%p with(numtheory): P:=proc(i) local a,j,k,n,s; d:=2; for n from 3 to i do k:=0; j:=n;

%p while j mod 2 <> 1 do k:=k+1; j:=j/2; od; a:=sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2;

%p d:=d+1/n^a; od; print(evalf(d, 300)); end: P(100);

%t f[n_] := Total@ Cases[Range[2, n - 1], _?(Abs[Mod[n, #] - #/2] < 1 &)]; First@ RealDigits@ N[Sum[1/k^f@ k, {k, 120}], 86] (* _Michael De Vlieger_, Oct 08 2015 *)

%Y Cf. A066417, A192265.

%K nonn,cons

%O 1,1

%A _Paolo P. Lava_, Jun 27 2011

%E Corrected and edited by _R. J. Mathar_, Jun 27 2011