A134681 Number of digits of all the divisors of n.

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

Offset: 1

Author

Reinhard Zumkeller, Nov 06 2007

Keywords

• nonn
• base

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

Crossrefs

Cf. A000005, A037278, A055642, A190997, A034690, A134682.

Programs

Maple

A134681 := proc(n)
    add(A055642(d),d=numtheory[divisors](n)) ;
end proc:
seq(A134681(n),n=1..80) ; # R. J. Mathar, Feb 21 2025

Mathematica

Array[Total[IntegerLength[Divisors[#]]]&,100] (* Harvey P. Dale, Jun 08 2013 *)

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

Formula

a(n) = A055642(A037278(n)) = Number of digits of the concatenation of all divisors of 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)

Extensions

New name from Jaroslav Krizek, Jun 15 2011