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A036537
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Numbers whose number of divisors is a power of 2.
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36
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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
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1,2
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COMMENTS
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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.
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
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)
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LINKS
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FORMULA
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EXAMPLE
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383, 384, 385, 386 have 2, 16, 8, 4 divisors, respectively, so they are consecutive terms of this sequence.
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MATHEMATICA
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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 *)
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PROG
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(Haskell)
a036537 n = a036537_list !! (n-1)
a036537_list = filter ((== 1) . a209229 . a000005) [1..]
(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)))
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CROSSREFS
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The closure of the squarefree numbers under application of A355038(.) and lcm.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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