OFFSET
1,2
COMMENTS
Also number of digits of the concatenation of all divisors of n (A037278). - Jaroslav Krizek, Jun 15 2011
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
FORMULA
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
KEYWORD
nonn,base
AUTHOR
Reinhard Zumkeller, Nov 06 2007
EXTENSIONS
New name from Jaroslav Krizek, Jun 15 2011
STATUS
approved