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A182627
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Total number of digits in binary expansion of all divisors of n.
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6
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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
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1,2
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COMMENTS
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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)
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LINKS
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FORMULA
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a(n) = tau(n) + Sum_{d|n} floor(log_2(d)). - Ridouane Oudra, Dec 11 2020
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EXAMPLE
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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.
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MATHEMATICA
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Table[Total[IntegerLength[Divisors[n], 2]], {n, 60}] (* Harvey P. Dale, Jan 26 2012 *)
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PROG
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(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))
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CROSSREFS
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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).
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KEYWORD
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nonn,base,easy
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AUTHOR
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STATUS
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approved
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