A175495 Positive integers k such that k < 2^d(k), where d(k) is the number of divisors of k.

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

Offset: 1

Author

Leroy Quet, May 30 2010

Keywords

• nonn

Comments

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.

All 52 terms of A034884 are also in this sequence. - Zak Seidov, May 30 2010

All powers of 2 are terms. - D. S. McNeil, May 30 2010

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

References

• 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.

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A175494, A034884.

Programs

Mathematica

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 *)

PARI

isok(n) = n < 2^numdiv(n); \\ Michel Marcus, Sep 09 2019

Python

from sympy import divisor_count
def ok(n): return n < 2**divisor_count(n)
print(list(filter(ok, range(1, 157)))) # Michael S. Branicky, Jul 29 2021

Extensions

More terms from Jon E. Schoenfield, Jun 13 2010