OFFSET
1,1
COMMENTS
a(n+1)/a(n) is very close to 2; a(n)/2^n is near 0.7.
As n goes to infinity, lim a(n)/2^n = 0.687827... (A327839; see comments in A036537). - Vladimir Shevelev, Feb 28 2017
FORMULA
a(n) = number of 1s in f(tau(k)) mapped across k = 1..2^n, with f(x):= 1-sign(log_2 x - floor( log_2 x )). - Michael De Vlieger, Mar 01 2017
EXAMPLE
Of the numbers 1 .. 2^4 = 16, only 4, 9, 12 and 16 are not in A036537, so a(4) = 16 - 4 = 12.
MAPLE
IversonBrackets := expr -> subs(true=1, false=0, expr):
A := proc(n) option remember; {seq(2^k, k=0..n)} end:
h := proc(n) option remember; add(evalb(numtheory:-tau(j) in A(n)), j=2^(n-1) + 1..2^n); IversonBrackets(%) end:
a := n -> 1 + add(h(k), k=1..n); seq(a(n), n=1..17); # Peter Luschny, May 14 2018
MATHEMATICA
Table[Count[#, 1] &@ Table[1 - Sign[# - Floor@ #] &@Log[2, #] &@ DivisorSigma[0, x], {x, 1, 2^m}], {m, 1, 20}] (* original program edited by Michael De Vlieger, Mar 01 2017, or *)
1 + Accumulate@ Table[Count[Range[2^(n - 1) + 1, 2^n], k_ /; IntegerQ@ Log2@ DivisorSigma[0, k]], {n, 20}] (* Michael De Vlieger, Feb 28 2017 *)
PROG
(PARI) a(n) = sum(k=1, 2^n, d = numdiv(k); (d<=2) || (ispower(d, , &p) && (p==2))); \\ Michel Marcus, May 14 2018
(Python)
from sympy import factorint
def A036538(n): return sum(1 for m in range(1, (1<<n)+1) if all(map(lambda w:not((k:=w+1)&-k)^k, factorint(m).values()))) # Chai Wah Wu, Jun 22 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
a(20)-a(26) from Michael De Vlieger, Feb 28 2017
a(27)-a(34) from Giovanni Resta, May 14 2018
STATUS
approved