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A025528
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Number of prime powers <= n with exponents > 0 (A246655).
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36
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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
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OFFSET
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1,3
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COMMENTS
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a(n) is the sum of the exponents in the prime factorization of lcm{1,2,...,n}.
Larger than but analogous to Pi(n).
Equally, number of finite fields of order <= n. - Neven Juric, Feb 05 2010
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REFERENCES
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G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, p. 203, Publications de l'Institut Cartan, 1990.
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LINKS
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FORMULA
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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) = 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)
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EXAMPLE
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Below 100 there are 25 primes and 25 + 10 = 35 prime powers.
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MATHEMATICA
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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 *)
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PROG
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(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))
(SageMath)
def A025528(n) : return sum([1 for k in (0..n) if is_prime_power(k)])
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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