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%I #41 Jul 30 2021 06:19:32
%S 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,
%T 54,56,60,63,64,66,70,72,78,80,84,88,90,96,100,102,104,105,108,110,
%U 112,114,120,126,128,130,132,135,136,138,140,144,150,152,154,156
%N Positive integers k such that k < 2^d(k), where d(k) is the number of divisors of k.
%C Numbers k for which A175494(k) = 1.
%C After the initial 1 in this sequence, the first integer in this sequence but not in A034884 is 44.
%C All 52 terms of A034884 are also in this sequence. - _Zak Seidov_, May 30 2010
%C All powers of 2 are terms. - _D. S. McNeil_, May 30 2010
%C 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
%D K. Prachar, Primzahlverteilung, Springer-Verlag, 1957, Ch. 1, Theorem 5.2.
%D S. Ramanujan, Highly composite numbers, Collected papers, Cambridge, 1927, 85-86.
%D 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.
%H T. D. Noe, <a href="/A175495/b175495.txt">Table of n, a(n) for n = 1..10000</a>
%t t = {}; n = 0; While[Length[t] < 100, n++; If[n < 2^DivisorSigma[0, n], AppendTo[t, n]]]; t (* _T. D. Noe_, May 14 2013 *)
%t Select[Range[200],#<2^DivisorSigma[0,#]&] (* _Harvey P. Dale_, Apr 24 2015 *)
%o (PARI) isok(n) = n < 2^numdiv(n); \\ _Michel Marcus_, Sep 09 2019
%o (Python)
%o from sympy import divisor_count
%o def ok(n): return n < 2**divisor_count(n)
%o print(list(filter(ok, range(1, 157)))) # _Michael S. Branicky_, Jul 29 2021
%Y Cf. A000005, A175494, A034884.
%K nonn
%O 1,2
%A _Leroy Quet_, May 30 2010
%E More terms from _Jon E. Schoenfield_, Jun 13 2010
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