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 A025528 Number of prime powers <= n with exponents > 0 (A246655). 27
 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) is the 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. 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 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) 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 (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 CROSSREFS Cf. A000961, A000040, A000720, A001221, A003418, A141228, A246655, A276781 (ordinal transform). One less than A065515. Sequence in context: A116549 A268382 A107079 * A255338 A123580 A072894 Adjacent sequences:  A025525 A025526 A025527 * A025529 A025530 A025531 KEYWORD nonn AUTHOR EXTENSIONS New description from Labos Elemer, Nov 09 2000 STATUS approved

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Last modified August 10 06:08 EDT 2022. Contains 356029 sequences. (Running on oeis4.)