OFFSET
1,2
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)
LINKS
Jaroslav Krizek, Table of n, a(n) for n = 1..500
FORMULA
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
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))
print([a(n) for n in range(1, 72)]) # Michael S. Branicky, Apr 21 2022
CROSSREFS
KEYWORD
nonn,base,easy
AUTHOR
Omar E. Pol, Nov 23 2010
STATUS
approved