A065515 Number of prime powers <= n.
1, 2, 3, 4, 5, 5, 6, 7, 8, 8, 9, 9, 10, 10, 10, 11, 12, 12, 13, 13, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 19, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 22, 22, 22, 22, 23, 23, 24, 24, 24, 24, 25, 25, 25, 25, 25, 25, 26, 26, 27, 27, 27, 28, 28, 28, 29, 29, 29, 29, 30, 30, 31
Offset: 1
Examples
There are 9 prime powers <= 12: 1=2^0, 2, 3, 4=2^2, 5, 7, 8=2^3, 9=3^2 and 11, so a(12) = 9.
References
- F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, Chapter 4.
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
- Eric Weisstein's World of Mathematics, Prime Power
Crossrefs
Programs
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Haskell
a065515 n = length $ takeWhile (<= n) a000961_list -- Reinhard Zumkeller, Apr 25 2011
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Maple
N:= 100: # to get a(1) to a(N) L:= Vector(N): L[1]:= 1: p:= 1: while p < N do p:= nextprime(p); for k from 1 to floor(log[p](N)) do L[p^k] := 1; od od: ListTools:-PartialSums(convert(L,list)); # Robert Israel, May 03 2015 -
Mathematica
a[n_] := 1 + Count[ Range[2, n], p_ /; Length[ FactorInteger[p]] == 1]; Table[a[n], {n, 1, 73}] (* Jean-François Alcover, Oct 12 2011 *) Accumulate[Table[If[Length[FactorInteger[n]]==1,1,0],{n,80}]] (* Harvey P. Dale, Aug 06 2016 *) Accumulate[Table[If[PrimePowerQ[n],1,0],{n,120}]]+1 (* Harvey P. Dale, Sep 29 2016 *) -
PARI
a(n)=n+=.5;1+sum(k=1,log(n)\log(2),primepi(n^(1/k))) \\ Charles R Greathouse IV, Apr 26 2012
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Python
from sympy import primepi from sympy.ntheory.primetest import integer_nthroot def A065515(n): return 1+sum(primepi(integer_nthroot(n,k)[0]) for k in range(1,n.bit_length())) # Chai Wah Wu, Jul 23 2024
Formula
Partial sums of A010055. - Reinhard Zumkeller, Nov 22 2009
a(n) = 1 + Sum_{k=1..log_2(n)} pi(floor(n^(1/k))). - Chayim Lowen, Aug 05 2015
a(n) = 1 + Sum_{k=2..n} floor(2*A001222(k)/(tau(k^2)-1)) where tau is A000005(n). - Anthony Browne, May 17 2016
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