cp's OEIS Frontend

A069637 Number of prime powers <= n with exponents > 1.

%I A069637 #38 Aug 16 2024 08:36:56
%S A069637 0,0,0,1,1,1,1,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,6,6,6,6,6,7,7,7,
%T A069637 7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,
%U A069637 9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10
%N A069637 Number of prime powers <= n with exponents > 1.
%C A069637 Counts A025475 without 1 = prime^0: a(n) = A085501(n) - 1. - _Reinhard Zumkeller_, Jul 03 2003
%C A069637 Counts the prime powers (A246655) without the primes. - _Peter Luschny_, Nov 18 2019
%D A069637 H. Sahu, K. Kar and B.S.K.R. Somayajulu, On the average order of pi*(n) - pi(n), Acta Cienc. Indica Math., Vol. 11 (1985), pp. 165-168.
%D A069637 József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter VII, p. 237.
%H A069637 Daniel Forgues, <a href="/A069637/b069637.txt">Table of n, a(n) for n=1..100000</a>.
%F A069637 a(n) = A025528(n) - A000720(n) = A000720([n^(1/2)]) + A000720([n^(1/3)]) + ... . - _Max Alekseyev_, May 11 2009
%F A069637 Sum_{k=1..n} a(k) ~ (4/3) * n^(3/2)/log(n) + O(n^(3/2)/log(n)^2) (Sahu et al., 1985). - _Amiram Eldar_, Mar 07 2021
%p A069637 with(numtheory);
%p A069637 A069637 := proc(N) local ct,i; ct:=0;
%p A069637 for i from 1 to N do if not isprime(i) and nops(factorset(i))=1 then ct:=ct+1; fi; od; ct; end; # _N. J. A. Sloane_, Jun 05 2022
%t A069637 Table[Sum[PrimePi[n^(1/k)], {k, Log[2, n]}]-PrimePi[n],{n,94}] (* _Stefano Spezia_, Jun 05 2022 *)
%o A069637 (SageMath)
%o A069637 [A025528(n) - prime_pi(n)  for n in (1..100)] # _Peter Luschny_, Nov 18 2019
%o A069637 (Python)
%o A069637 from sympy import primepi, integer_nthroot
%o A069637 def A069637(n): return sum(primepi(integer_nthroot(n,k)[0]) for k in range(2,n.bit_length())) # _Chai Wah Wu_, Aug 15 2024
%Y A069637 Cf. A025528, A000720, A025475, A085501, A246655.
%Y A069637 Partial sums of A268340.
%K A069637 nonn
%O A069637 1,8
%A A069637 _Amarnath Murthy_, Mar 27 2002