cp's OEIS Frontend

A089580 Total number of perfect powers > 1 below 10^n, counting multiple representations separately.

%I A089580 #71 Mar 09 2025 12:52:28
%S A089580 3,15,49,143,406,1174,3507,10674,32965,102716,321797,1011533,3186389,
%T A089580 10050743,31730134,100228040,316713623,1001037546,3164497349,
%U A089580 10004755374,31632975598,100021893194,316274794666,1000101078148,3162495003352,10000467510247,31623782520064,100002164895587
%N A089580 Total number of perfect powers > 1 below 10^n, counting multiple representations separately.
%C A089580 From _Robert G. Wilson v_, Jul 17 2016: (Start)
%C A089580 a(n) ~ sqrt(10^n).
%C A089580 a(n) - A089579(n) = A275358(n).
%C A089580 The four terms which make up the difference a(2) - A089579(2) are: 16 = 2^4 = 4^2, 64 = 2^6 = 4^3 = 8^2 and 81 = 3^4 = 9^2; one for 16, two for 64 and one for 81 making a total of 4. See A117453.
%C A089580 (End)
%C A089580 This sequence correlates (see Link) to A006880 via a power fit A*x^B. For example, using a(23) through a(29) one obtains (A,B) = (0.047272, 1.96592) with R^2 > 0.999999. This extrapolates A006880(30) as 1.46*10^28. The exponent well may be resolving to 2. - _Bill McEachen_, Mar 04 2025
%H A089580 Chai Wah Wu, <a href="/A089580/b089580.txt">Table of n, a(n) for n = 1..1998</a> (n = 1..100 from Robert G. Wilson, n = 101..400 from Karl-Heinz Hofmann)
%H A089580 Karl-Heinz Hofmann, <a href="/A089580/a089580.txt">Python program</a>.
%H A089580 Bill McEachen, <a href="/A089580/a089580.png">Plot A089580 vs A006880</a>.
%F A089580 a(n) = Sum_{k = 1..n} A060298(k). - _Karl-Heinz Hofmann_, Sep 18 2023
%e A089580 16 = 2^4 = 4^2 counts double, 256 = 2^8 = 4^4 = 16^2 counts three times.
%t A089580 Table[lim=10^n-1; Sum[Floor[lim^(1/k)]-1, {k,2,Floor[Log[2,lim]]}], {n,30}] (* _T. D. Noe_, Nov 16 2006 *)
%o A089580 (Python) # see link.
%Y A089580 Cf. A001597, A072103, A117453, A275358, A060298.
%Y A089580 Cf. A089579 (counting multiple representations only once).
%K A089580 nonn
%O A089580 1,1
%A A089580 _Martin Renner_, Dec 29 2003
%E A089580 2 more terms from _Martin Renner_, Oct 02 2004
%E A089580 More terms from _T. D. Noe_, Nov 16 2006
%E A089580 More precise name by _Hugo Pfoertner_, Sep 16 2023