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

KEYWORD

nonn

AUTHOR

Leroy Quet, May 30 2010

EXTENSIONS

More terms from Jon E. Schoenfield, Jun 13 2010

STATUS

approved