

A076411


Number of perfect powers < n.


4



0, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12
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OFFSET

1,5


COMMENTS

Perfect powers are in A001597. The function a(n) increases much more slowly than pi(n): e.g., a(1765)=54 and pi(1765)=274. See also A076412.
a(n) >= A000196(n1).  Robert Israel, Jul 31 2015
This is essentially the same as A069623 which is the main entry, see there for more formulas.  M. F. Hasler, Aug 16 2015


LINKS

Harvey P. Dale, Table of n, a(n) for n = 1..1000


FORMULA

a(n) = n^(1/2) + n^(1/3) + n^(1/5)  n^(1/6) + n^(1/7)  n^(1/10) + O(n^(1/11)).  Charles R Greathouse IV, Aug 14 2015


EXAMPLE

a(9)=3 because there are 3 perfect powers less than 9: 1,4,8.


MATHEMATICA

Join[{0}, Accumulate[Table[If[GCD@@FactorInteger[n][[All, 2]]>1, 1, 0], {n, 90}]]+1] (* Harvey P. Dale, Mar 19 2020 *)


PROG

(PARI) a(n)=n; nsum(k=1, logint(n, 2), moebius(k)*(sqrtnint(n, k)1)) \\ Charles R Greathouse IV, Jul 21 2017


CROSSREFS

A069623(n) = a(n+1) is the main entry.
Cf. A001597, A076412, A075802, A096623.
Sequence in context: A234451 A085501 A069623 * A217038 A309196 A243283
Adjacent sequences: A076408 A076409 A076410 * A076412 A076413 A076414


KEYWORD

nonn


AUTHOR

Zak Seidov, Oct 09 2002


STATUS

approved



