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

3, 15, 49, 143, 406, 1174, 3507, 10674, 32965, 102716, 321797, 1011533, 3186389, 10050743, 31730134, 100228040, 316713623, 1001037546, 3164497349, 10004755374, 31632975598, 100021893194, 316274794666, 1000101078148, 3162495003352, 10000467510247, 31623782520064, 100002164895587

Offset: 1

Author

Martin Renner, Dec 29 2003

Keywords

• nonn

Comments

From Robert G. Wilson v, Jul 17 2016: (Start)

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

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

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.

(End)

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

Examples

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

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..1998 (n = 1..100 from Robert G. Wilson, n = 101..400 from Karl-Heinz Hofmann)
• Karl-Heinz Hofmann, Python program.
• Bill McEachen, Plot A089580 vs A006880.

Crossrefs

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

Cf. A089579 (counting multiple representations only once).

Programs

Mathematica

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 *)

Python

# see link.

Formula

a(n) = Sum_{k = 1..n} A060298(k). - Karl-Heinz Hofmann, Sep 18 2023

Extensions

2 more terms from Martin Renner, Oct 02 2004

More terms from T. D. Noe, Nov 16 2006

More precise name by Hugo Pfoertner, Sep 16 2023