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A025528 Number of prime powers <= n with exponents > 0. 13
0, 1, 2, 3, 4, 4, 5, 6, 7, 7, 8, 8, 9, 9, 9, 10, 11, 11, 12, 12, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 18, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 21, 21, 21, 21, 22, 22, 23, 23, 23, 23, 24, 24, 24, 24, 24, 24, 25, 25, 26, 26, 26, 27, 27, 27, 28, 28, 28, 28, 29, 29, 30, 30 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

a(n) = sum of the exponents in the prime factorization of lcm{1,2,...,n}.

Larger than but analogous to Pi(n).

Counts A000961 without 1=prime^0: a(n)=A065515(n)-1. - Reinhard Zumkeller, Jul 03 2003

Equally, number of finite fields of order <= n. - Neven Juric, Feb 05 2010

REFERENCES

G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, p. 203, Publications de l'Institut Cartan, 1990.

LINKS

Daniel Forgues, Table of n, a(n) for n = 1..100000.

Index entries for sequences related to lcm's

FORMULA

a(n) = Cardinality[{1..n}|A001221(i)=1].

a(n) = Sum_{p prime <= n} floor(log(n)/log(p)). - Benoit Cloitre, Apr 30 2002

a(n) ~ n/log(n). - Benoit Cloitre, May 30 2003

a(n) = A069637(n) + A000720(n). - Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Feb 24 2004 [Corrected by Franklin T. Adams-Watters, Jun 08 2008]

a(n) = A000720(n) + A000720(floor(n^(1/2))) + A000720(floor(n^(1/3))) + ... - Max Alekseyev, May 11 2009

Partial sums of A069513. - Enrique Pérez Herrero, May 30 2011

a(n) = A001222(A003418(n)). - Luc Rousseau, Jan 05 2018

EXAMPLE

Below 100 there are 25 primes and 25+10 = 35 prime powers.

MATHEMATICA

primePowerPi[n_] := Sum[PrimePi[n^(1/k)], {k, Log[2, n]}]; Table[primePowerPi[n], {n, 75}] (* Geoffrey Critzer, Jan 07 2012 *) (* and modified by Robert G. Wilson v, Jan 07 2012 *)

Table[Sum[Boole[1 < Cyclotomic[n, 1]], {n, 1, m}], {m, 1, 75}] (* Fred Daniel Kline, Oct 03 2016 *)

PROG

(PARI) for(n=1, 100, print1(sum(k=1, n, logint(n, prime(k))), ", ")) \\ corrected by Luc Rousseau, Jan 04 2018

(PARI) a(n)=sum(i=1, n, if(omega(i)-1, 0, 1))

(PARI) a(n)=n+=.5; sum(e=1, log(n)\log(2), primepi(n^(1/e))) \\ Charles R Greathouse IV, Apr 30 2012

CROSSREFS

Cf. A000961, A000040, A000720, A001221, A141228.

Sequence in context: A116549 A268382 A107079 * A255338 A123580 A072894

Adjacent sequences:  A025525 A025526 A025527 * A025529 A025530 A025531

KEYWORD

nonn

AUTHOR

Clark Kimberling

EXTENSIONS

New description from Labos Elemer, Nov 09 2000

STATUS

approved

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Last modified November 13 13:15 EST 2018. Contains 317149 sequences. (Running on oeis4.)