cp's OEIS Frontend

A385971 Smallest m such that 5^m begins with n 9's after the first digit.

%I A385971 #35 Jul 22 2025 23:05:04
%S A385971 0,8,195,799,28737,167821,325146,6432162,543157237,1807789217,
%T A385971 3731189547,3731189547
%N A385971 Smallest m such that 5^m begins with n 9's after the first digit.
%C A385971 a(n) is also the smallest m such that 1/2^m begins with n 9's after the first nonzero digit.
%C A385971 a(n) is equal to A152561(n)-1 for n=2, 6, 7 and possibly for many other terms.
%C A385971 When summing a series with dominant term 1/2^m (such as the Riemann zeta function), the n 9's here show how small further terms must be to avoid changing the initial decimal digit from 1/2^m.
%H A385971 Marco Ripà, <a href="https://math.stackexchange.com/q/5079681">Is the leading digit of the decimal expansion of the prime zeta function at n equal to the first digit of 5^n, for all integers n≥10?</a>, Stack Exchange (2025)
%F A385971 a(n) = Min_{d=1..9} S(d*10^(n+1)-1) where 5^S(k) is the smallest power of 5 beginning with k.
%e A385971 5^a(0) = 5^0      = 1
%e A385971 5^a(1) = 5^8      = 390625
%e A385971 5^a(2) = 5^195    = 1991364888915565346...
%e A385971 5^a(3) = 5^799    = 2999393627791261909...
%e A385971 5^a(4) = 5^28737  = 1999929120817815105...
%e A385971 5^a(5) = 5^167821 = 6999994116858573262...
%Y A385971 Cf. A152561, A000351, A111395, A011557.
%K A385971 base,nonn,hard,more
%O A385971 0,2
%A A385971 _Giulio Bonfissuto_, Jul 13 2025