This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A134681 #43 Feb 21 2025 05:32:47
%S A134681 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 A134681 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 A134681 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 A134681 Number of digits of all the divisors of n.
%C A134681 Also number of digits of the concatenation of all divisors of n (A037278). - _Jaroslav Krizek_, Jun 15 2011
%H A134681 Reinhard Zumkeller, <a href="/A134681/b134681.txt">Table of n, a(n) for n = 1..10000</a>
%F A134681 a(n) = A055642(A037278(n)) = Number of digits of the concatenation of all divisors of n.
%F A134681 From _Sida Li_, Sep 01 2023: (Start)
%F A134681 a(n) = Sum_{d divides n} (floor(log_10(d))+1).
%F A134681 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 A134681 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 A134681 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)
%p A134681 A134681 := proc(n)
%p A134681 add(A055642(d),d=numtheory[divisors](n)) ;
%p A134681 end proc:
%p A134681 seq(A134681(n),n=1..80) ; # _R. J. Mathar_, Feb 21 2025
%t A134681 Array[Total[IntegerLength[Divisors[#]]]&,100] (* _Harvey P. Dale_, Jun 08 2013 *)
%o A134681 (PARI) a(n) = sumdiv(n, d, #digits(d)); \\ _Michel Marcus_, Sep 01 2023
%o A134681 (Python)
%o A134681 from sympy import divisors
%o A134681 def a(n): return sum(len(str(d)) for d in divisors(n, generator=True))
%o A134681 print([a(n) for n in range(1, 97)]) # _Michael S. Branicky_, Nov 03 2023
%Y A134681 Cf. A000005, A037278, A055642, A190997, A034690, A134682.
%K A134681 nonn,base
%O A134681 1,2
%A A134681 _Reinhard Zumkeller_, Nov 06 2007
%E A134681 New name from _Jaroslav Krizek_, Jun 15 2011