A285261 a(2n-1) and a(2n) represent the smallest fraction a(2n-1)/a(2n) needing n unit fractions.

1, 2, 2, 3, 4, 5, 8, 11, 16, 17, 77, 79, 732, 733, 27538, 27539

Offset: 1

Author

Robert G. Wilson v, Apr 15 2017

Keywords

• hard
• more
• nonn

Comments

Huang Zhibin of China in April 2014 has verified that this fraction (27538/27539) needs 8 unit fractions.

Examples

			For n=1, 1/2 = 1/2;
for n=2, 2/3 = 1/3 + 1/3;
for n=3, 4/5 = 1/2 + 1/4 + 1/20 and 4/5 = 1/2 + 1/5 + 1/10;
for n=4, 8/11 = 1/2 + 1/6 + 1/22 + 1/66;
for n=5, 16/17 = 1/2 + 1/3 + 1/17 + 1/34 + 1/51;
for n=6, 77/79 = 1/2 + 1/3 + 1/8 + 1/79 + 1/474 + 1/632;
for n=7, 732/733 = 1/2 + 1/3 + 1/7 + 1/45 + 1/7330 + 1/20524 + 1/26388;
for n=8, 27538/27539 = 1/2 + 1/3 + 1/7 + 1/43 + 1/1933 + 1/14893663 + 1/1927145066572824 + 1/212829231672162931784; etc.
		

Links

• Ron Knott, Egyptian Fractions

Crossrefs

Cf. A097048, A097049.

Formula

a(2n-1) = A097049(n); a(2n) = A097048(n). - Jon E. Schoenfield, Jan 11 2020