login
Least number of fractions of the form (k+1)/k, for k a positive integer, whose product equals n.
3

%I #13 Sep 11 2024 23:02:07

%S 0,1,2,2,3,3,4,3,4,4,5,4,5,5,5,4,5,5,6,5,6,6,7,5,6,6,6,6,7,6,7,5,6,6,

%T 7,6,7,7,7,6,7,7,7,7,7,8,8,6,7,7,7,7,8,7,8,7,8,8,9,7,8,8,8,6,7,7,8,7,

%U 8,8,9,7,8,8,8,8,8,8,9,7,8,8,9,8,8,8,9,8

%N Least number of fractions of the form (k+1)/k, for k a positive integer, whose product equals n.

%C If each intermediate product of the first j of the fractions, for all j < a(n), is also restricted to be an integer, the resulting sequence is A117497. The first n for which a shorter product can be obtained by allowing intermediate non-integer products is 43 = 2/1 * 2/1 * 2/1 * 2/1 * 2/1 * 4/3 * 129/128, a product of 7 fractions, where A117497(43) = 8.

%H Joseph Myers, <a href="/A277608/b277608.txt">Table of n, a(n) for n = 1..10000</a>

%H United Kingdom Mathematics Trust, <a href="https://bmos.ukmt.org.uk/home/ukmog-2016.pdf">Mathematical Olympiad for Girls 2016</a>, problem 5.

%Y Cf. A117497 (restriction to intermediate products being integers), A014701 (always generating n from n-1 for n odd and from n/2 for n even), A376012.

%K nonn

%O 1,3

%A _Joseph Myers_, Oct 23 2016