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A070428
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Number of perfect powers (A001597) not exceeding 10^n.
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13
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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
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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FORMULA
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a(n) ~ sqrt(10^n).
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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
(Python)
from sympy import mobius, integer_nthroot
def A070428(n): return int(1-sum(mobius(x)*(integer_nthroot(10**n, x)[0]-1) for x in range(2, (10**n).bit_length()))) # Chai Wah Wu, Aug 13 2024
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CROSSREFS
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Cf. A089579, A089580 (number of perfect powers (not including 1) < 10^n).
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KEYWORD
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easy,nonn,changed
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AUTHOR
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Oct 03 2002
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STATUS
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approved
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