A294651 Least possible value for the highest denominator in the decomposition of unity as a sum of different unitary fractions the greatest of which is 1/n.

1, 6, 15, 20, 24, 28, 33, 40, 48, 52, 65, 65, 75, 76, 85, 88, 91, 100, 105, 115, 115, 119, 132, 140, 144, 145, 155, 161, 162, 171, 217, 174, 182, 190, 195, 196, 296, 200, 207, 220, 246, 224, 301, 231, 238, 253, 329, 275, 280, 287, 288, 296, 371, 300, 304, 305

Offset: 1

Author

Javier Múgica, Nov 06 2017

Keywords

• nonn
• nice

Comments

The decompositions need not be unique. E.g., for a(7) either 1/12 or 1/20 + 1/30 may be used in the decomposition indifferently.

For prime numbers p and any fixed epsilon < 1, a(p) > epsilon*p*log(p) for all sufficiently large p.

Examples

			1 = 1/3 + 1/4 + 1/6 + 1/10 + 1/12 + 1/15, and there is no such decomposition starting at 1/3 and having a greatest denominator smaller than 15, so a(3)=15.
		

Links

• Jon E. Schoenfield, All unitary decompositions (listed in lexicographic order) for n = 1..30. (Decompositions up to n = 18 originally found by J. Múgica.)
• Javier Múgica, Values of a(n)/n.

Crossrefs

Cf. A192881, which looks at decompositions with the least possible number of terms. Those from this sequence achieve those bounds up to a(7), with exception of a(3). However, n=7 is likely the last value of n for which this holds.

Cf. A272083.

Extensions

a(18)-a(24) from Jon E. Schoenfield, Dec 22 2019

a(25)-a(56) from Jon E. Schoenfield, Jan 01 2020