

A182627


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


6



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, 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, 7, 28, 14, 28, 14, 14, 7, 42, 7, 14, 21, 28, 15, 30, 8, 24, 15, 30, 8
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OFFSET

1,2


COMMENTS

Also, total number of digits in row n of triangle A182620.
Also, number of digits of A182621(n).
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.
1 2 2 3 3 3 3 4 4 4 4 4 4 4 4 5 ...
0 1 0 2 0 2 0 3 0 3 0 3 0 3 0 4 ...
0 0 1 0 0 2 0 0 2 0 0 3 0 0 3 0 ...
0 0 0 1 0 0 0 2 0 0 0 2 0 0 0 3 ...
0 0 0 0 1 0 0 0 0 2 0 0 0 0 2 0 ...
0 0 0 0 0 1 0 0 0 0 0 2 0 0 0 0 ...
0 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 ...
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 2 ...
0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 ...
0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 ...
0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 ...
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 ...
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 ...
0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 ...
0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 ...
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ...
...

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


LINKS



FORMULA

a(n) = tau(n) + Sum_{dn} floor(log_2(d)).  Ridouane Oudra, Dec 11 2020


EXAMPLE

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.


MATHEMATICA

Table[Total[IntegerLength[Divisors[n], 2]], {n, 60}] (* Harvey P. Dale, Jan 26 2012 *)


PROG

(PARI) a(n) = sumdiv(n, d, 1+logint(d, 2)); \\ Michel Marcus, Dec 11 2020
(Python)
from sympy import divisors
def a(n): return sum(d.bit_length() for d in divisors(n))


CROSSREFS

Cf. A093653 (number of 1's in binary expansion of all divisors of n).
Cf. A226590 (number of 0's in binary expansion of all divisors of n).


KEYWORD

nonn,base,easy


AUTHOR



STATUS

approved



