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Number of digits of all the divisors of n.
5

%I #40 Dec 15 2023 15:05:27

%S 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,

%T 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,

%U 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

%N Number of digits of all the divisors of n.

%C Also number of digits of the concatenation of all divisors of n (A037278). - _Jaroslav Krizek_, Jun 15 2011

%H Reinhard Zumkeller, <a href="/A134681/b134681.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A055642(A037278(n)).

%F From _Sida Li_, Sep 01 2023: (Start)

%F a(n) = Sum_{d divides n} (floor(log_10(d))+1).

%F 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).

%F 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.

%F 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)

%t Array[Total[IntegerLength[Divisors[#]]]&,100] (* _Harvey P. Dale_, Jun 08 2013 *)

%o (PARI) a(n) = sumdiv(n, d, #digits(d)); \\ _Michel Marcus_, Sep 01 2023

%o (Python)

%o from sympy import divisors

%o def a(n): return sum(len(str(d)) for d in divisors(n, generator=True))

%o print([a(n) for n in range(1, 97)]) # _Michael S. Branicky_, Nov 03 2023

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

%K nonn,base

%O 1,2

%A _Reinhard Zumkeller_, Nov 06 2007

%E New name from _Jaroslav Krizek_, Jun 15 2011