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A386758 Number of decimal digits in the n-th Lucas number.

%I A386758 #32 Aug 21 2025 17:45:06
%S A386758 1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,6,6,7,7,7,7,7,
%T A386758 8,8,8,8,8,9,9,9,9,9,10,10,10,10,11,11,11,11,11,12,12,12,12,12,13,13,
%U A386758 13,13,13,14,14,14,14,15,15,15,15,15,16,16,16,16,16,17,17,17,17,17
%N A386758 Number of decimal digits in the n-th Lucas number.
%C A386758 As F(n)<=L(n), the number of decimal digits of the Lucas number L(n) is at least as large as the number of decimal digits of the Fibonacci number F(n). Furthermore, the difference is at most one. The indices for which the difference is one is A386760.
%H A386758 Hans J. H. Tuenter, <a href="/A386758/b386758.txt">Table of n, a(n) for n = 0..10000</a>
%F A386758 a(n) = A055642(A000032(n)).
%F A386758 a(n) = 1 + floor(n*log_10(phi)), where log_10(phi) = A097348, and phi = (1+sqrt(5))/2 = A001622.
%e A386758 L(0)=2 has one digit, so that a(0)=1; L(5)=11 has two digits, so that a(5)=2.
%p A386758 a:= n-> 1+floor(n*log[10]((1+sqrt(5))/2)):
%p A386758 seq(a(n), n=0..81);
%t A386758 a[n_] := IntegerLength[LucasL[n]]; Array[a, 100, 0] (* _Amiram Eldar_, Aug 16 2025 *)
%Y A386758 Cf. A000032, A001622, A055642, A060384, A097348.
%Y A386758 Number of digits of L(p^n): A094057 (p=2), A114469 (p=10).
%K A386758 base,nonn,easy,new
%O A386758 0,6
%A A386758 _Hans J. H. Tuenter_, Aug 06 2025