cp's OEIS Frontend

A211665 Minimal number of iterations of log_10 applied to n until the result is < 1.

%I A211665 #28 May 14 2025 09:08:47
%S A211665 1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,
%T A211665 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,
%U A211665 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2
%N A211665 Minimal number of iterations of log_10 applied to n until the result is < 1.
%C A211665 Different from A055642 and A138902, cf. Example.
%C A211665 Instead the real-valued log function one can consider only the integer part (i.e., A004216), since log_b(x) < k <=> x < b^k <=> floor(x) < b^k for any integer k >= 0; that's also why the first 2, 3, 4, ... appears exactly for 10, 10^10, 10^(10^10) etc. - _M. F. Hasler_, Dec 12 2018
%F A211665 With E_{i=1..n} c(i) := c(1)^(c(2)^(c(3)^(...(c(n-1)^(c(n)))...))); E_{i=1..0} := 1; example: E_{i=1..3} 10 = 10^(10^10) = 10^10000000000, we have:
%F A211665 a(E_{i=1..n} 10) = a(E_{i=1..n-1} 10) + 1, for n >= 1.
%F A211665 G.f.: g(x) = (1/(1-x))*Sum_{k>=0} x^(E_{i=1..k} 10) = (x + x^10 + x^(10^10) + ...)/(1-x).
%e A211665 a(n) = 1, 2, 3, 4 for n = 1, 10, 10^10, 10^(10^10), i.e., n = 1, 10, 10000000000, 10^10000000000.
%e A211665 a(n) = 2 for all n >= 10, n < 10^10.
%t A211665 a[n_] := Length[NestWhileList[Log10, n, # >= 1 &]] - 1; Array[a, 100] (* _Amiram Eldar_, Dec 08 2018 *)
%o A211665 (PARI) a(n,i=1)={while(n=logint(n,10),i++);i} \\ _M. F. Hasler_, Dec 07 2018
%Y A211665 Cf. A001069, A010096, A211661, A211663, A211666, A211668, A211670.
%K A211665 base,nonn
%O A211665 1,10
%A A211665 _Hieronymus Fischer_, Apr 30 2012
%E A211665 Name reworded by _M. F. Hasler_, Dec 12 2018