A074738 Decimal expansion of d = 1-(1+log(log(2)))/log(2) = 0.08607133....

0, 8, 6, 0, 7, 1, 3, 3, 2, 0, 5, 5, 9, 3, 4, 2, 0, 6, 8, 8, 7, 5, 7, 3, 0, 9, 8, 7, 7, 6, 9, 2, 2, 6, 7, 7, 7, 6, 0, 5, 9, 1, 1, 0, 9, 5, 3, 0, 3, 3, 3, 1, 7, 3, 4, 9, 2, 0, 2, 0, 2, 3, 6, 6, 6, 5, 4, 2, 2, 6, 3, 5, 8, 1, 4, 6, 2, 2, 8, 7, 9, 7, 9, 9, 3, 8, 0, 5, 3, 4, 6, 0, 2, 5, 2, 8, 7, 6, 8, 0, 7, 1, 6, 3

Offset: 0

Author

Benoit Cloitre, Sep 05 2002

Keywords

• cons
• easy
• nonn

Comments

An Erdős constant: let s(x) denotes the number of numbers < x expressible as a product of 2 numbers less than or equal to sqrt(x). Erdős showed that S(x) is x/(log x)^(d+o(1)) where d is this constant.

Ford finds that, if H(x,y,z) is the number of integers n <= x which have a divisor in the interval (y,z] and for 3 <= y <= sqrt(x), H(x,y,2y) = x/(((log y)^delta)(log log y)^(3/2)) where delta is the Erdős constant whose decimal digits are A074738. - Jonathan Vos Post, Jul 19 2007

Occurs, citing Ford, in p.2 of Koukoulopoulos. - Jonathan Vos Post, May 18 2010

Luca & Pomerance call this the Erdős-Tenenbaum-Ford constant and show its relationship to the reduced totient function A002174. - Charles R Greathouse IV, Dec 28 2013

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Kevin Ford, Integers with a divisor in (y,2y], arXiv:math/0607473 [math.NT], 2006-2013.
• Andrew Granville, Cihan Sabuncu, and Alisa Sedunova, The multiplication table constant and sums of two squares, arXiv:2308.14911 [math.NT], 2023.
• Dimitris Koukoulopoulos, Divisors of shifted primes, arXiv:0905.0163 [math.NT], 2009-2010; International Mathematics Research Notices, 2010:24, pp. 4585-4627.
• Florian Luca and Carl Pomerance, On the range of Carmichael's universal-exponent function, Acta Arithmetica 162 (2014), pp. 289-308.
• G. Tenenbaum, Sur la probabilité qu'un entier possède un diviseur dans un intervalle donné, Compositio Mathematica, 51 no. 2 (1984), p. 243-263 (see Theorem 1).

Programs

Magma

1-(1+Log(Log(2)))/Log(2); // G. C. Greubel, Apr 16 2018

Maple

evalf(1-(1+log(log(2)))/log(2), 119);  # Alois P. Heinz, Aug 30 2023

Mathematica

Join[{0}, RealDigits[1 - (1 + Log[Log[2]])/Log[2], 10, 100][[1]]] (* G. C. Greubel, Apr 16 2018 *)

PARI

1-(1+log(log(2)))/log(2) \\ Michel Marcus, Mar 14 2013