OFFSET
1,1
COMMENTS
This sequence was the subject of the 1st problem of the 3rd Benelux Mathematical Olympiad in 2011, where a pair (k, m) is called a 'Benelux pair' (see links).
Every pair (2^q-2, 2^q*(2^q-2)) for q >= 2 is a solution, the next such pairs are (4094, 16769024), (8190, 67092480), (16382, 268402688), (32766, 1073676288), ... hence there exist infinitely many Benelux pairs.
Only one pair is known to be not of this form (75, 1215) (see examples).
LINKS
Thomas Bloom, Problem 850, Erdős Problems.
BxMO 2011, Third Benelux Mathematical Olympiad, Problème 1.
Diophante, A1853. Deux miniatures bénéluxiennes (in French).
Erdős problems database contributors, Erdős problem database, see no. 850.
Christian Hercher, On one of Erdős' Problems - An Efficient Search for Benelux Pairs, arXiv:2506.01099 [math.NT], 2025. See p. 13.
EXAMPLE
First pairs are (2, 8), (6, 48), (14, 224), (30, 960), (75, 1215), (62, 3968), (126, 16128), ...
Examples corresponding to solutions (2^q-2, 2^q*(2^q-2)):
-> For q = 2, a(1) = 2 = 2^1 and a(2) = 8 = 2^3 while 3 = 3^1 and 9 = 3^2.
-> For q = 3, a(3) = 6 = 2 * 3 and a(4) = 48 = 2^4 * 3 while 7 = 7^1 and 49 = 7^2.
The only known solution not of that form: a(9) = 75 = 3 * 5^2 and a(10) = 1215 = 5 * 3^5 while 76 = 2^2 * 19 and 1216 = 2^6 * 19.
CROSSREFS
KEYWORD
nonn,tabf,hard,more
AUTHOR
Bernard Schott, Apr 05 2021
EXTENSIONS
Confirmed a(23)-a(30) and extended with a(31)-a(32) by Martin Ehrenstein, Apr 18 2021
STATUS
approved