Problem 144

The density of integers which have two divisors $d_1,d_2$ such that $d_1<d_2<2d_1$ exists and is equal to $1$ .

Status

SolvedProved

An accepted solution exists. The statement is true.

PROVED (LEAN).

Er64h: Erdős, P., On some applications of probability to analysis and number theory. J. London Math. Soc. (1964), 692-696.
Er79: Erdős, Paul, Some unconventional problems in number theory. Math. Mag. (1979), 67-70.
ErHa79: Erdős, P. and Hall, R. R., The propinquity of divisors. Bull. London Math. Soc. (1979), 304-307.
Gu04: Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp.; doi:10.1007/978-0-387-26677-0. Section E3 "Density of integers with two comparable divisors", printed p. 313, which poses the question and reports the Maier and Tenenbaum solution. Library home: Guy 2004.
MaTe84: Maier, H. and Tenenbaum, G., On the set of divisors of an integer. Invent. Math. (1984), 121-128.