A025528 Number of prime powers <= n with exponents > 0 (A246655).
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
Offset: 1
Keywords
Examples
Below 100 there are 25 primes and 25 + 10 = 35 prime powers.
References
- G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, p. 203, Publications de l'Institut Cartan, 1990.
Links
Crossrefs
Programs
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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 *) -
PARI
for(n=1,100,print1(sum(k=1,n,logint(n,prime(k))),",")) \\ corrected by Luc Rousseau, Jan 04 2018
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PARI
a(n)=sum(i=1,n,if(omega(i)-1,0,1))
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PARI
a(n)=n+=.5;sum(e=1,log(n)\log(2),primepi(n^(1/e))) \\ Charles R Greathouse IV, Apr 30 2012
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Python
from sympy import primepi, integer_nthroot def A025528(n): return sum(primepi(integer_nthroot(n,k)[0]) for k in range(1,n.bit_length())) # Chai Wah Wu, Aug 15 2024
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SageMath
def A025528(n) : return sum([1 for k in (0..n) if is_prime_power(k)]) print([A025528(n) for n in (1..74)]) # Peter Luschny, Nov 18 2019
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
From Steven Foster Clark, Sep 26 2018: (Start)
a(n) = Sum_{m=1..n} A001222(m) * A002321(floor(n/m)) where A001222() is the Omega function and A002321() is the Mertens function.
a(n) = Sum_{m=1..floor(log_2(n))} A000010(m)/m * J(floor(n^(1/m))) where A000010() is Euler's totient function and J(n) = Sum_{m=1..floor(log_2(n))} 1/m * A000720(floor(n^(1/m))) is Riemann's prime-power counting function.
(End)
Extensions
New description from Labos Elemer, Nov 09 2000
Comments