%I #20 Oct 13 2021 19:15:00
%S 1,3,8,20,50,119,280,645,1466,3280,7262,15937,34720,75108,161552,
%T 345785,736974,1564762,3311206,6985780,14698342,30850276,64607782,
%U 135030018,281689074,586636098,1219788256,2532608855,5251282902,10874696106,22493653324,46475828418
%N a(n) = Sum_{k=1..2^n} d(k) where d(n) = number of divisors of n (A000005).
%H Chai Wah Wu, <a href="/A085831/b085831.txt">Table of n, a(n) for n = 0..82</a> (terms 0..64 from Donovan Johnson)
%t k = s = 0; Do[ While[ k < 2^n, k++; s = s + DivisorSigma[ 0, k ]]; Print[s], {n, 0, 29} ]
%o (Python)
%o from math import isqrt
%o def A085831(n): return (lambda m, r: 2*sum(r//k for k in range(1, m+1))-m*m)(isqrt(2**n),2**n) # _Chai Wah Wu_, Oct 08 2021
%o (PARI) a(n) = sum(k=1, 2^n, numdiv(k)); \\ _Michel Marcus_, Oct 10 2021
%Y Cf. A006218, A050226, A085567, A085829, A057494.
%K nonn
%O 0,2
%A _Robert G. Wilson v_, Sep 21 2000