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

Author

Labos Elemer

Keywords

• nonn

Examples

			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.
		

Crossrefs

Cf. A036537, A037992, A327839.

Programs

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 *)

PARI

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

Formula

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

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