A182627 Total number of digits in binary expansion of all divisors of n.

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

Offset: 1

Author

Omar E. Pol, Nov 23 2010

Keywords

• nonn
• base
• easy

Comments

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

Also, number of digits of A182621(n).

Rows sums of triangle A182628.

From Davide Rotondo, Apr 20 2022: (Start)

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 ...

...

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

Examples

			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.
		

Links

• Jaroslav Krizek, Table of n, a(n) for n = 1..500

Crossrefs

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

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).

Programs

Mathematica

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

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))
print([a(n) for n in range(1, 72)]) # Michael S. Branicky, Apr 21 2022

Formula

a(n) = A093653(n) + A226590(n). - Jaroslav Krizek, Sep 01 2013

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

a(n) = Sum_{i=0..floor(log_2(n))} A135539(n,2^i). - Ridouane Oudra, Sep 19 2022