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%I #29 Aug 19 2021 09:50:22
%S 0,1,2,4,7,9,13,16,20,26,31,40,50,61,78,93,119,150,189,242,310,400,
%T 525,684,900,1190,1581,2117,2836,3807,5136,6948,9425,12811,17437,
%U 23788,32517,44512,60971,83640,114899,157948,217336,299360,412635,569193,785753,1085319,1500140,2074794,2870849,3974425,5504966
%N Number of prime powers (p^2, p^3, ...) <= 2^n.
%H Ray Chandler, <a href="/A036386/b036386.txt">Table of n, a(n) for n = 1..125</a> (first 112 terms from Hiroaki Yamanouchi)
%H <a href="/index/Pri#primepop">Index entries for sequences related to numbers of primes in various ranges</a>
%F a(n) = Sum_{j=2..n+1} pi(floor(2^(n/j))). The summation starts with squares(j=2); for arbitrary range (=y), the y^(1/j) argument has to be used.
%e The 9 prime powers not exceeding 64 are 4, 8, 9, 16, 25, 27, 32, 49, 64.
%e n = 25, a(25) = 900, pi(5792) + pi(322) + pi(76) + pi(32) + pi(17) + pi(11) + pi(8) + pi(6) + pi(5) + pi(4) + pi(4) + pi(3) + pi(3) + pi(3) + pi(2) + pi(2) + pi(2) + pi(2) + pi(2) + pi(2) + pi(2) + pi(2) + pi(2) + pi(2) + pi(1) = 760 + 66 + 21 + 11 + 7 + 5 + 4 + 3 + 3 + 2 + 2 + 2 + 2 + 2 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 0.
%t f[n_] := Length@ Union@ Flatten@ Table[ Prime[j]^k, {k, 2, n + 1}, {j, PrimePi[2^(n/k)]}]; Array[f, 46] (* _Robert G. Wilson v_, Jul 08 2011 *)
%Y Cf. A007053, A029837, A036378-A036390.
%K nonn
%O 1,3
%A _Labos Elemer_
%E More terms from _Labos Elemer_, May 07 2001
%E Terms a(47) and beyond from _Hiroaki Yamanouchi_, Nov 15 2016