%I #46 Sep 19 2022 20:58:57

%S 1,3,3,6,4,8,4,10,7,10,5,15,5,10,10,15,6,17,6,18,11,12,6,24,9,12,12,

%T 18,6,24,6,21,13,14,13,30,7,14,13,28,7,26,7,21,20,14,7,35,10,21,14,21,

%U 7,28,14,28,14,14,7,42,7,14,21,28,15,30,8,24,15,30,8

%N Total number of digits in binary expansion of all divisors of n.

%C Also, total number of digits in row n of triangle A182620.

%C Also, number of digits of A182621(n).

%C Rows sums of triangle A182628.

%C From _Davide Rotondo_, Apr 20 2022: (start)

%C Can be constructed by writing the sequence of natural numbers with 1 one, 2 twos, 4 threes, 8 fours, ..., where 1,2,4,8,... are consecutive powers of 2; then the same sequence spaced by a zero, then the same sequence spaced by two zeros, and so on. Finally add the values of the columns.

%C 1 2 2 3 3 3 3 4 4 4 4 4 4 4 4 5 ...

%C 0 1 0 2 0 2 0 3 0 3 0 3 0 3 0 4 ...

%C 0 0 1 0 0 2 0 0 2 0 0 3 0 0 3 0 ...

%C 0 0 0 1 0 0 0 2 0 0 0 2 0 0 0 3 ...

%C 0 0 0 0 1 0 0 0 0 2 0 0 0 0 2 0 ...

%C 0 0 0 0 0 1 0 0 0 0 0 2 0 0 0 0 ...

%C 0 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 ...

%C 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 2 ...

%C 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 ...

%C 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 ...

%C 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ...

%C 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ...

%C 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 ...

%C 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 ...

%C 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 ...

%C 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ...

%C ...

%C ----------------------------------------------

%C Tot. 1 3 3 6 4 8 4 10 7 10 5 15 5 10 10 15 ... (End)

%H Jaroslav Krizek, <a href="/A182627/b182627.txt">Table of n, a(n) for n = 1..500</a>

%F a(n) = A093653(n) + A226590(n). - _Jaroslav Krizek_, Sep 01 2013

%F a(n) = tau(n) + Sum_{d|n} floor(log_2(d)). - _Ridouane Oudra_, Dec 11 2020

%F a(n) = Sum_{i=0..floor(log_2(n))} A135539(n,2^i). - _Ridouane Oudra_, Sep 19 2022

%e The divisors of 12 are 1, 2, 3, 4, 6, 12. These divisors written in base 2 are 1, 10, 11, 100, 110, 1100. Then a(12)=15 because 1+2+2+3+3+4 = 15.

%t Table[Total[IntegerLength[Divisors[n],2]],{n,60}] (* _Harvey P. Dale_, Jan 26 2012 *)

%o (PARI) a(n) = sumdiv(n, d, 1+logint(d, 2)); \\ _Michel Marcus_, Dec 11 2020

%o (Python)

%o from sympy import divisors

%o def a(n): return sum(d.bit_length() for d in divisors(n))

%o print([a(n) for n in range(1, 72)]) # _Michael S. Branicky_, Apr 21 2022

%Y Cf. A093653, A135539, A182620, A182621, A182628.

%Y Cf. A093653 (number of 1's in binary expansion of all divisors of n).

%Y Cf. A226590 (number of 0's in binary expansion of all divisors of n).

%K nonn,base,easy

%O 1,2

%A _Omar E. Pol_, Nov 23 2010