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

1,2

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

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 *)

Prog

(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

Crossrefs

Cf. A000005, A175494, A034884.

Sequence in context: A081000 A336367 A064377 * A034884 A137291 A032954

Adjacent sequences: A175492 A175493 A175494 * A175496 A175497 A175498

Keyword

nonn

Author

Leroy Quet, May 30 2010

Extensions

More terms from Jon E. Schoenfield, Jun 13 2010

Status

approved