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

1,2

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

R. 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

Crossrefs

Cf. A000040, A000961, A000720, A276781 (ordinal transform).

A025528(n) = a(n) - 1.

Cf. A139555. - Reinhard Zumkeller, Oct 27 2010

Sequence in context: A196165 A356990 A138366 * A070545 A254828 A091863

Adjacent sequences: A065512 A065513 A065514 * A065516 A065517 A065518

Keyword

nice,nonn

Author

Reinhard Zumkeller, Nov 27 2001

Status

approved