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A070428 Number of perfect powers (A001597) not exceeding 10^n. 13
1, 4, 13, 41, 125, 367, 1111, 3395, 10491, 32670, 102231, 320990, 1010196, 3184138, 10046921, 31723592, 100216745, 316694005, 1001003332, 3164437425, 10004650118, 31632790244, 100021566157, 316274216762, 1000100055684 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
In the programs for this sequence, 4*n can be replaced by the smaller floor(n*log(10)/log(2)). - T. D. Noe, Nov 17 2006
REFERENCES
The Dominion (Wellington, NZ), 'wtd sell', 9 Nov. 1991.
sci.math, powers not exceeding n. nz science monthly advt, March 1993, 1:80 integers 1..10000 is perfect square or higher power.
LINKS
Eric Weisstein's World of Mathematics, Perfect Power
FORMULA
a(n) ~ sqrt(10^n).
EXAMPLE
a(1) = 4 because the powers 1, 4, 8, 9 do not exceed 10^1.
a(2) = 13 because 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81 & 100, are the only perfect power numbers less than or equal to 100.
MATHEMATICA
f[n_] := 1 - Sum[ MoebiusMu[x]*Floor[10^(n/x) - 1], {x, 2, n*Log2[10]}]; Array[f, 25, 0] (* Robert G. Wilson v, May 22 2009; modified Aug 04 2014 *)
PROG
(PARI) for(n=0, 25, print1(sum(x=2, 4*n, -moebius(x)*(floor(10^(n/x)-1)))+1, ", ")); \\ Slightly modified by Jinyuan Wang, Mar 02 2020
CROSSREFS
Cf. A001597.
Cf. A089579, A089580 (number of perfect powers (not including 1) < 10^n).
Sequence in context: A213496 A220926 A077284 * A320563 A368344 A268989
KEYWORD
easy,nonn
AUTHOR
Donald S. McDonald, May 15 2002
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Oct 03 2002
Edited and extended by Robert G. Wilson v, Oct 11 2002
STATUS
approved

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Last modified March 19 03:33 EDT 2024. Contains 370952 sequences. (Running on oeis4.)