A036537 Numbers whose number of divisors is a power of 2.

1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 93, 94, 95, 97, 101, 102

Offset: 1

Author

Labos Elemer

Keywords

• nonn

Comments

Primes and A030513(d(x)=4) are subsets; d(16k+4) and d(16k+12) have the form 3Q, so x=16k+4 or 16k-4 numbers are missing.

A number m is a term if and only if all its divisors are infinitary, or A000005(m) = A037445(m). - Vladimir Shevelev, Feb 23 2017

All exponents in the prime number factorization of a(n) have the form 2^k-1, k >= 1. So it is an S-exponential sequence (see Shevelev link) with S={2^k-1}. Using Theorem 1, we obtain that a(n) ~ C*n, where C = Product((1-1/p)*(1 + Sum_{i>=1} 1/p^(2^i-1))). - Vladimir Shevelev Feb 27 2017

This constant is C = 0.687827... . - Peter J. C. Moses, Feb 27 2017

From Peter Munn, Jun 18 2022: (Start)

1 and numbers j*m^2, j squarefree, m >= 1, such that all prime divisors of m divide j, and m is in the sequence.

Equivalently, the nonempty set of numbers whose squarefree part (A007913) and squarefree kernel (A007947) are equal, and whose square part's square root (A000188) is in the set.

(End)

Examples

			383, 384, 385, 386 have 2, 16, 8, 4 divisors, respectively, so they are consecutive terms of this sequence.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Vladimir Shevelev, S-exponential numbers, Acta Arithm., 175(2016), 385-395.
• Eric Weisstein's World of Mathematics, Square Part, Squarefree Part

Crossrefs

A005117, A030513, A058891, A175496, A336591 are subsequences.

Complement of A162643; subsequence of A002035. - Reinhard Zumkeller, Jul 08 2009

Subsequence of A162644, A337533.

Cf. A000005, A000188, A007913, A007947, A036538, A352780.

The closure of the squarefree numbers under application of A355038(.) and lcm.

Programs

Haskell

a036537 n = a036537_list !! (n-1)
a036537_list = filter ((== 1) . a209229 . a000005) [1..]
-- Reinhard Zumkeller, Nov 15 2012

Mathematica

bi[ x_ ] := 1-Sign[ N[ Log[ 2, x ], 5 ]-Floor[ N[ Log[ 2, x ], 5 ] ] ]; ld[ x_ ] := Length[ Divisors[ x ] ]; Flatten[ Position[ Table[ bi[ ld[ x ] ], {x, 1, m} ], 1 ] ]
Select[Range[110],IntegerQ[Log[2,DivisorSigma[0,#]]]&] (* Harvey P. Dale, Nov 20 2016 *)

PARI

is(n)=n=numdiv(n);n>>valuation(n,2)==1 \\ Charles R Greathouse IV, Mar 27 2013

PARI

isok(m) = issquarefree(m) || (omega(m) == omega(core(m)) && isok(core(m,1)[2])); \\ Peter Munn, Jun 18 2022

Python

from itertools import count, islice
from sympy import factorint
def A036537_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:all(map(lambda m:not((k:=m+1)&-k)^k,factorint(n).values())),count(max(startvalue,1)))
A036537_list = list(islice(A036537_gen(),30)) # Chai Wah Wu, Jan 04 2023

Formula

A209229(A000005(a(n))) = 1. - Reinhard Zumkeller, Nov 15 2012

a(n) << n. - Charles R Greathouse IV, Feb 25 2017

m is in the sequence iff for k >= 0, A352780(m, k+1) | A352780(m, k)^2. - Peter Munn, Jun 18 2022