A036539 a(n) is the number of numbers k with 2^(n-1) < k <= 2^n having a number of divisors that is a power of 2.

1, 1, 4, 5, 11, 22, 44, 89, 178, 351, 702, 1413, 2817, 5634, 11273, 22542, 45077, 90150, 180322, 360621, 721233, 1442482, 2884968, 5769917, 11539863, 23079674, 46159310, 92318616, 184637146, 369274400, 738548882, 1477097703, 2954195153, 5908390134, 11816780283

Offset

1,3

Links

Table of n, a(n) for n=1..35.

Formula

a(n) ~ c * 2^(n-1), where c = 0.687827... (A327839). - Amiram Eldar, Aug 16 2024

Example

a(5) = 11: The following 11 numbers have numbers of divisors that are powers of 2: 17, 19, 21, 22, 23, 24, 26, 27, 29, 30 and 31 with 2, 2, 4, 4, 2, 8, 4, 4, 2, 8 and 2 divisors, respectively.

Mathematica

f[n_] := Boole[n == 2^IntegerExponent[n, 2]]; a[n_] := Sum[f[DivisorSigma[0, k]], {k, 2^(n - 1) + 1, 2^n}]; Array[a, 20] (* Amiram Eldar, Aug 16 2024 *)

Prog

(PARI) a(n)=sum(k=2^(n-1)+1, 2^n, my(d=numdiv(k)); (d/(1<<valuation(d, 2)))==1 ); \\ Joerg Arndt, Feb 27 2017

Crossrefs

Cf. A036537, A037992, A327839.

Sequence in context: A185507 A000286 A227620 * A279710 A303956 A000769

Adjacent sequences: A036536 A036537 A036538 * A036540 A036541 A036542

Keyword

nonn

Author

Labos Elemer

Extensions

Name clarified and more terms from Joerg Arndt, Feb 27 2017

a(25)-a(28) from Jon E. Schoenfield, Jul 31 2018

a(29)-a(35) from Jon E. Schoenfield, Aug 04 2018

Status

approved