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

1,1

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

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.

Formula

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

Example

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

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 *)

Prog

(Python) # see link.

Crossrefs

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

Cf. A089579 (counting multiple representations only once).

Sequence in context: A316853 A061316 A124161 * A034564 A012203 A112586

Adjacent sequences: A089577 A089578 A089579 * A089581 A089582 A089583

Keyword

nonn,changed

Author

Martin Renner, Dec 29 2003

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

Status

approved