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%I #64 Dec 12 2023 08:24:45
%S 1,2,4,7,11,19,28,45,71,118,199,341,605,1079,1962,3591,6636,12371,
%T 23151,43580,82268,155922,296348,564689,1078556,2064590,3959000,
%U 7605135,14632961,28195587,54403836,105102702,203287170,393625232,762951923,1480223717,2874422304
%N Position of 2^n among the powers of primes (A000961).
%C Number of prime powers <= 2^n. - _Jon E. Schoenfield_, Nov 06 2016
%C A000961(a(n)) = A000079(n); also position of record values in A192015: A001787(n) = A192015(a(n)). - _Reinhard Zumkeller_, Jun 26 2011
%H Ray Chandler, <a href="/A024622/b024622.txt">Table of n, a(n) for n = 0..92</a> (using b-file from A007053, first 61 terms from Hiroaki Yamanouchi)
%F From _Ridouane Oudra_, Oct 26 2020: (Start)
%F a(n) = 1 + Sum_{i=1..n} pi(floor(2^(n/i))), where pi(n) = A000720(n);
%F a(n) = 1 + A182908(n). (End)
%t {1}~Join~Flatten[1 + Position[Select[Range[10^6], PrimePowerQ], k_ /; IntegerQ@ Log2@ k ]] (* _Michael De Vlieger_, Nov 14 2016 *)
%o (PARI) lista(nn) = {v = vector(2^nn, i, i); vpp = select(x->ispp(x), v); print1(1, ", "); for (i=1, #vpp, if ((vpp[i] % 2) == 0, print1(i, ", ")););} \\ _Michel Marcus_, Nov 17 2014
%o (PARI) a(n)=sum(k=1,n,primepi(sqrtnint(2^n,k)))+1 \\ _Charles R Greathouse IV_, Nov 21 2014
%o (PARI) a(n)=my(s=0);for(i=1, 2^n, isprimepower(i) && s++);s+1 \\ _Dana Jacobsen_, Mar 23 2021
%o (SageMath) def a(n): return sum(prime_pi(ZZ(2^n).nth_root(k+1,truncate_mode=1)[0]) for k in range(n))+1 # _Dana Jacobsen_, Mar 23 2021
%o (Perl) use ntheory ":all"; for my $n (0..20) { my $s=1; is_prime_power($_) && $s++ for 1..2**$n; print "$n $s\n" } # _Dana Jacobsen_, Mar 23 2021
%o (Perl) use ntheory ":all"; for my $n (0..64) { my $s = ($n < 1) ? 1 : vecsum(map{prime_count(rootint(powint(2,$n)-1,$_))}1..$n)+2; print "$n $s\n"; } # _Dana Jacobsen_, Mar 23 2021
%o (Perl) # with b-file for pi(2^n)
%o perl -Mntheory=:all -nE 'my($n,$pc)=split; say "$n ", addint($pc,vecsum( map{prime_count(rootint(powint(2,$n),$_))} 2..$n )+1);' b007053.txt # _Dana Jacobsen_, Mar 23 2021
%Y Cf. A000720, A000961, A007053, A182908.
%K nonn
%O 0,2
%A _Clark Kimberling_
%E a(28)-a(36) from _Hiroaki Yamanouchi_, Nov 21 2014
%E a(46)-a(53) corrected by _Hiroaki Yamanouchi_, Nov 15 2016
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