

A036537


Numbers whose number of divisors is a power of 2.


31



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
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OFFSET

1,2


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 16k4 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^k1, k >= 1. So it is an Sexponential sequence (see Shevelev link) with S={2^k1}. Using Theorem 1, we obtain that a(n) ~ C*n, where C = Product((11/p)*(1 + Sum_{i>=1} 1/p^(2^i1))).  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)


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Vladimir Shevelev, Sexponential numbers, Acta Arithm., 175(2016), 385395.
Eric Weisstein's World of Mathematics, Square Part, Squarefree Part


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


EXAMPLE

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


MATHEMATICA

bi[ x_ ] := 1Sign[ 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 *)


PROG

(Haskell)
a036537 n = a036537_list !! (n1)
a036537_list = filter ((== 1) . a209229 . a000005) [1..]
 Reinhard Zumkeller, Nov 15 2012
(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


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.
Sequence in context: A268335 A002035 A336591 * A072510 A084116 A137620
Adjacent sequences: A036534 A036535 A036536 * A036538 A036539 A036540


KEYWORD

nonn


AUTHOR

Labos Elemer


STATUS

approved



