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Total number of perfect powers > 1 below 10^n, counting multiple representations separately.
6

%I #59 Oct 16 2023 16:07:19

%S 3,15,49,143,406,1174,3507,10674,32965,102716,321797,1011533,3186389,

%T 10050743,31730134,100228040,316713623,1001037546,3164497349,

%U 10004755374,31632975598,100021893194,316274794666,1000101078148,3162495003352,10000467510247,31623782520064,100002164895587

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

%C From _Robert G. Wilson v_, Jul 17 2016: (Start)

%C a(n) ~ sqrt(10^n).

%C a(n) - A089579(n) = A275358(n).

%C 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 (End)

%H 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 Karl-Heinz Hofmann, <a href="/A089580/a089580.txt">Python program.</a>

%F a(n) = Sum_{k = 1..n} A060298(k). - _Karl-Heinz Hofmann_, Sep 18 2023

%e 16 = 2^4 = 4^2 counts double, 256 = 2^8 = 4^4 = 16^2 counts three times.

%t 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 (Python) # see link.

%Y Cf. A001597, A072103, A117453, A275358, A060298.

%Y Cf. A089579 (counting multiple representations only once).

%K nonn

%O 1,1

%A _Martin Renner_, Dec 29 2003

%E 2 more terms from _Martin Renner_, Oct 02 2004

%E More terms from _T. D. Noe_, Nov 16 2006

%E More precise name by _Hugo Pfoertner_, Sep 16 2023