OFFSET
1,2
COMMENTS
a(n) > pi(n) = A000720(n).
From Chayim Lowen, Aug 05 2015: (Start)
a(n) <= pi(n) + A069623(n).
Conjecture: a(n) >= pi(A069623(n)) + pi(n) + 1.
Each term m is repeated A057820(m) times. (End)
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
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
EXAMPLE
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.
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 *)
PROG
(Haskell)
a065515 n = length $ takeWhile (<= n) a000961_list
-- Reinhard Zumkeller, Apr 25 2011
(PARI) a(n)=n+=.5; 1+sum(k=1, log(n)\log(2), primepi(n^(1/k))) \\ Charles R Greathouse IV, Apr 26 2012
(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
CROSSREFS
KEYWORD
nice,nonn
AUTHOR
Reinhard Zumkeller, Nov 27 2001
STATUS
approved