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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.
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%I #29 Aug 16 2024 16:42:07

%S 1,1,4,5,11,22,44,89,178,351,702,1413,2817,5634,11273,22542,45077,

%T 90150,180322,360621,721233,1442482,2884968,5769917,11539863,23079674,

%U 46159310,92318616,184637146,369274400,738548882,1477097703,2954195153,5908390134,11816780283

%N 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.

%F a(n) ~ c * 2^(n-1), where c = 0.687827... (A327839). - _Amiram Eldar_, Aug 16 2024

%e 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.

%t 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 *)

%o (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

%Y Cf. A036537, A037992, A327839.

%K nonn

%O 1,3

%A _Labos Elemer_

%E Name clarified and more terms from _Joerg Arndt_, Feb 27 2017

%E a(25)-a(28) from _Jon E. Schoenfield_, Jul 31 2018

%E a(29)-a(35) from _Jon E. Schoenfield_, Aug 04 2018