A192266 Decimal expansion of Sum_{k >= 1} 1/k^sigma_*(k) where sigma_*(n) is the sum of the anti-divisors of n.
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, 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, 7, 1, 8, 2, 8, 4, 3, 0, 3, 2, 7, 1, 4, 2, 5, 6, 4, 8
Offset: 1
Examples
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..
Programs
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Maple
with(numtheory): P:=proc(i) local a,j,k,n,s; d:=2; for n from 3 to i do k:=0; j:=n; 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; d:=d+1/n^a; od; print(evalf(d, 300)); end: P(100);
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Mathematica
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 *)
Extensions
Corrected and edited by R. J. Mathar, Jun 27 2011
Comments