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A175495
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Positive integers k such that k < 2^d(k), where d(k) is the number of divisors of k.
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15
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1, 2, 3, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 105, 108, 110, 112, 114, 120, 126, 128, 130, 132, 135, 136, 138, 140, 144, 150, 152, 154, 156
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OFFSET
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1,2
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COMMENTS
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Numbers k for which A175494(k) = 1.
After the initial 1 in this sequence, the first integer in this sequence but not in A034884 is 44.
It follows from the Wiman-Ramanujan theorem that, for every eps > 0 and k > k_0(eps), we have k > tau(k)^(log(log(k))/(log(2)+eps)). Therefore in particular A034884 is finite. On the other hand, for 0 < eps < log(2), it is known that there exist infinitely many numbers for which k < tau(k)^(log(log(k))/(log(2)-eps)), that is, tau(k) > k^((log(2)-eps)/log(log(k))) and 2^tau(k) > 2^(k^((log(2)-eps)/log(log(k)))) >> k. In particular, A175495 is infinite. - Vladimir Shevelev, May 30 2010
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REFERENCES
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K. Prachar, Primzahlverteilung, Springer-Verlag, 1957, Ch. 1, Theorem 5.2.
S. Ramanujan, Highly composite numbers, Collected papers, Cambridge, 1927, 85-86.
A. Wiman, Sur l'ordre de grandeur du nombre de diviseurs d'un entier, Arkiv Mat. Astr. och Fys., 3, no. 18 (1907), 1-9.
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LINKS
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MATHEMATICA
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t = {}; n = 0; While[Length[t] < 100, n++; If[n < 2^DivisorSigma[0, n], AppendTo[t, n]]]; t (* T. D. Noe, May 14 2013 *)
Select[Range[200], #<2^DivisorSigma[0, #]&] (* Harvey P. Dale, Apr 24 2015 *)
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PROG
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(Python)
from sympy import divisor_count
def ok(n): return n < 2**divisor_count(n)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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