A036761 Number of refactorable integers (A033950) of binary order (A029837) n.

1, 1, 0, 1, 2, 2, 4, 8, 13, 22, 39, 77, 137, 254, 459, 889, 1665, 3175, 6041, 11619, 22319, 42979, 83123, 160649, 311826, 605225, 1176998, 2291702, 4466923, 8716126, 17023771, 33279942, 65109458, 127484313, 249783733, 489738130, 960801221, 1886039740

Offset: 0

Author

Labos Elemer

Keywords

• nonn

Comments

Since for any epsilon d(n) <= n^epsilon if n is large enough, a(n) does not grow very quickly.

Examples

			{1} has binary order 0, {2} has binary order 1, no term has binary order 2, {8} has binary order 3, {9,12} have binary order 4, {18,24} have binary order 5, ...
The 8 numbers, between 65 and 128 (with binary order 7) which are divided by d(x) (A000005) are 72,80,84,88,96,104,108,128, so a(7)=8.
		

Crossrefs

Cf. A000005, A029837, A033950.

Programs

Maple

with(numtheory): A036761 := proc(n) local ct,k,lim: if(n=0)then return 1: else ct:=0: lim:=2^n: for k from 2^(n-1)+1 to lim do if(k mod tau(k) = 0)then ct:=ct+1: fi: od: return ct: fi: end: seq(A036761(n),n=0..10); # Nathaniel Johnston, May 04 2011

Mathematica

Table[Count[Range[2^(n - 1) + 1, 2^(n)], k_ /; Divisible[k, DivisorSigma[0, k]]] + Boole[n == 0], {n, 0, 22}] (* Michael De Vlieger, May 20 2017 *)

Extensions

a(22)-a(37) from Donovan Johnson, Aug 29 2012