A024622 Position of 2^n among the powers of primes (A000961).

1, 2, 4, 7, 11, 19, 28, 45, 71, 118, 199, 341, 605, 1079, 1962, 3591, 6636, 12371, 23151, 43580, 82268, 155922, 296348, 564689, 1078556, 2064590, 3959000, 7605135, 14632961, 28195587, 54403836, 105102702, 203287170, 393625232, 762951923, 1480223717, 2874422304

Offset

0,2

Comments

Number of prime powers <= 2^n. - Jon E. Schoenfield, Nov 06 2016

A000961(a(n)) = A000079(n); also position of record values in A192015: A001787(n) = A192015(a(n)). - Reinhard Zumkeller, Jun 26 2011

Links

Ray Chandler, Table of n, a(n) for n = 0..92 (using b-file from A007053, first 61 terms from Hiroaki Yamanouchi)

Formula

From Ridouane Oudra, Oct 26 2020: (Start)

a(n) = 1 + Sum_{i=1..n} pi(floor(2^(n/i))), where pi(n) = A000720(n);

a(n) = 1 + A182908(n). (End)

Mathematica

{1}~Join~Flatten[1 + Position[Select[Range[10^6], PrimePowerQ], k_ /; IntegerQ@ Log2@ k ]] (* Michael De Vlieger, Nov 14 2016 *)

Prog

(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
(PARI) a(n)=sum(k=1, n, primepi(sqrtnint(2^n, k)))+1 \\ Charles R Greathouse IV, Nov 21 2014
(PARI) a(n)=my(s=0); for(i=1, 2^n, isprimepower(i) && s++); s+1 \\ Dana Jacobsen, Mar 23 2021
(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
(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
(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 $s\n"; } # Dana Jacobsen, Mar 23 2021
(Perl) # with b-file for pi(2^n)
perl -Mntheory=:all -nE 'my($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

Crossrefs

Cf. A000720, A000961, A007053, A182908.

Sequence in context: A326471 A326596 A170804 * A034337 A083024 A003292

Adjacent sequences: A024619 A024620 A024621 * A024623 A024624 A024625

Keyword

nonn

Author

Clark Kimberling

Extensions

a(28)-a(36) from Hiroaki Yamanouchi, Nov 21 2014

a(46)-a(53) corrected by Hiroaki Yamanouchi, Nov 15 2016

Status

approved