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A134681 Number of digits of all the divisors of n. 5
1, 2, 2, 3, 2, 4, 2, 4, 3, 5, 3, 7, 3, 5, 5, 6, 3, 7, 3, 8, 5, 6, 3, 10, 4, 6, 5, 8, 3, 11, 3, 8, 6, 6, 5, 12, 3, 6, 6, 11, 3, 11, 3, 9, 8, 6, 3, 14, 4, 9, 6, 9, 3, 11, 6, 11, 6, 6, 3, 18, 3, 6, 8, 10, 6, 12, 3, 9, 6, 12, 3, 17, 3, 6, 9, 9, 6, 12, 3, 15, 7, 6, 3, 18, 6, 6, 6, 12, 3, 18, 6, 9, 6, 6, 6, 18 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Also number of digits of the concatenation of all divisors of n (A037278). - Jaroslav Krizek, Jun 15 2011
LINKS
FORMULA
a(n) = A055642(A037278(n)).
From Sida Li, Sep 01 2023: (Start)
a(n) = Sum_{d divides n} (floor(log_10(d))+1).
log_10(Product_{d divides n} d) <= a(n) <= log_10(Product_{d divides n} d) + sigma_0(n), where sigma_0(n) = A000005(n).
Equivalently, sigma_0(n)*log_10(n)/2 <= a(n) <= sigma_0(n)*log_10(n)/2 + sigma_0(n), obtained by formula in A007955.
For x >= 5, c2*log(x)^2 + c1*log(x) + c0 <= (1/x)*Sum_{n<=x} a(n) <= c2*log(x)^2 + (c1+1)*log(x) + 2*c0, where c2 = 1/(2*log(10)), c1 = (gamma-1)/log(10), c0 = 2*gamma-1, and gamma is Euler's constant. This is obtained by hyperbola trick for Sum_{n<=x} sigma_0(n), and Abel partial summation on Sum_{n<=x} sigma_0(n)*log(n). (End)
MATHEMATICA
Array[Total[IntegerLength[Divisors[#]]]&, 100] (* Harvey P. Dale, Jun 08 2013 *)
PROG
(PARI) a(n) = sumdiv(n, d, #digits(d)); \\ Michel Marcus, Sep 01 2023
(Python)
from sympy import divisors
def a(n): return sum(len(str(d)) for d in divisors(n, generator=True))
print([a(n) for n in range(1, 97)]) # Michael S. Branicky, Nov 03 2023
CROSSREFS
Sequence in context: A324105 A328871 A169819 * A218703 A326641 A144372
KEYWORD
nonn,base
AUTHOR
Reinhard Zumkeller, Nov 06 2007
EXTENSIONS
New name from Jaroslav Krizek, Jun 15 2011
STATUS
approved

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Last modified April 18 22:18 EDT 2024. Contains 371782 sequences. (Running on oeis4.)