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A060298 Number of powers x^y (x,y > 1) with n digits.

%I A060298 #67 Oct 16 2023 17:43:55
%S A060298 3,12,34,94,263,768,2333,7167,22291,69751,219081,689736,2174856,
%T A060298 6864354,21679391,68497906,216485583,684323923,2163459803,6840258025,
%U A060298 21628220224,68388917596,216252901472,683826283482,2162393925204,6837972506895,21623315009817
%N A060298 Number of powers x^y (x,y > 1) with n digits.
%C A060298 Conjectures from _Robert G. Wilson v_, Aug 29 2012: (Start)
%C A060298 Limit_{n->oo} a(2n)/10^n = 1 - 1/sqrt(10).
%C A060298 Limit_{n->oo} a(2n-1)/10^n = 1/sqrt(10) - 1/10. (End)
%C A060298 These follow from the Formula. - _Robert Israel_, Apr 29 2020
%C A060298 Limit_{n->oo} a(n)/a(n-1) = sqrt(10). - _Bernard Schott_, Jan 21 2023
%H A060298 Robert Israel, <a href="/A060298/b060298.txt">Table of n, a(n) for n = 1..1000</a>
%H A060298 Karl-Heinz Hofmann, <a href="/A060298/a060298.txt">Python program</a>
%F A060298 a(n) = Sum_{y=2..floor(n*log_2(10))} (ceiling(10^(n/y)) - ceiling(10^((n-1)/y))) for n >= 2. - _Robert Israel_, Apr 29 2020
%F A060298 a(n) = A089580(n+1) - A089580(n) for n > 1. - _Karl-Heinz Hofmann_, Sep 18 2023
%e A060298 a(1) = 3 because there are 3 powers with 1 digit: 2^2, 2^3 and 3^2.
%p A060298 f:= proc(n) local y;
%p A060298   add(ceil(10^(n/y))-ceil(10^((n-1)/y)), y=2..floor(n*log[2](10)))
%p A060298 end proc:
%p A060298 f(1):= 3:
%p A060298 map(f, [$1..20]); # _Robert Israel_, Apr 29 2020
%o A060298 (Python) # see link
%o A060298 (Python)
%o A060298 from sympy import integer_nthroot, integer_log
%o A060298 def A060298(n):
%o A060298     if n == 1: return 3
%o A060298     c, y, a, b, t = 0, 2, 10**n-1, 10**(n-1)-1, (10**n).bit_length()
%o A060298     while y<t:
%o A060298         c += (m:=integer_nthroot(a,y)[0])-(k:=integer_nthroot(b,y)[0])
%o A060298         y = (integer_log(b,k)[0] if m==k else y)+1
%o A060298     return c # _Chai Wah Wu_, Oct 16 2023
%Y A060298 Cf. A001597, A089580 (partial sums).
%K A060298 nonn,base
%O A060298 1,1
%A A060298 _Michel ten Voorde_, Apr 10 2001
%E A060298 a(10)-a(18) from _Donovan Johnson_, Dec 14 2009
%E A060298 a(19)-a(27) from _Donovan Johnson_, Aug 29 2012