A006585 -id:A006585 - cp's OEIS frontend

A357765 Smallest positive integer that can be represented as the sum of n of its (possibly equal) divisors in the maximum number of ways (=A002966(n)).

1, 2, 12, 2520, 48348686786400, 10543141534556403817127800577537146514577188497111149855093902265479066128013109211427715400552367011213513440000

Offset: 1

David A. Corneth and Max Alekseyev, Oct 12 2022

			a(3) = 12 since for n = 3 the tuples forming the solutions of 1 = 1/x_1 + 1/x_2 + 1/x_3 are (x_1, x_2, x_3) in {(2, 3, 6), (2, 4, 4), (3, 3, 3)}. All these terms combined have an lcm of 12. The ways to write 12 as a sum of 3 of its divisors are therefore 12 = 12/2 + 12/3 + 12/6 = 6 + 4 + 2. Similarily we have 12 = 6 + 3 + 3 and 12 = 4 + 4 + 4.

Max Alekseyev, Table of n, a(n) for n = 1..7

Apparently coincides with A181700 for n >= 4.

Cf. A002966, A006585.

a(n) = LCM of all denominators of Egyptian fractions enumerated by A002966(n).

A144063 Egyptian fractions: number of solutions of 1 = 1/x_1 + ... + 1/x_n in positive integers x_1 < ... < x_n <= 256.

Original entry on oeis.org

1, 0, 1, 6, 62, 642, 5623, 47126, 368680, 2715613, 18876751, 124137535, 774232619, 4595291801, 26030660449, 141031079451, 731862267491, 3641135367129, 17379359388167, 79633646141291, 350541406992141, 1483638948734104, 6043258919626951, 23713645892145709

Offset: 1

Zhao Hui Du, Sep 09 2008

Egyptian fraction for a rational number is to represent the number in sum of some distinct unit fraction, such as 1 = 1/2 + 1/3 + 1/6, here we represent 1 by a three terms Egyptian fraction and the largest denominator is 6. And the representation is non-unique.

There are a total of 3007198863516917545589795267613 Egyptian fractions for 1 whose largest denominator is no more than 256.

			a(1)=1 since 1 = 1/1.
a(2)=0 since the sum of any two distinct unit fractions are not 1.
a(3)=1 since the only three terms Egyptian fraction for 1 is 1/2 + 1/3 + 1/6.

Jinyuan Wang, Table of n, a(n) for n = 1..114
Eric Weisstein's World of Mathematics, Egyptian Fraction
link for the result [Broken link]

Cf. A002967, A006585.

a(n) = 0 for n > 114.

A379452 Number of compositions (ordered partitions) of 1 into n distinct reciprocals of positive integers.

Original entry on oeis.org

1, 0, 6, 144, 8640, 1670400, 1238655600, 6095673521280

Offset: 1

Ilya Gutkovskiy, Dec 23 2024

			a(3) = 6 because we have 1 = 1/2 + 1/3 + 1/6
                           = 1/2 + 1/6 + 1/3
                           = 1/3 + 1/2 + 1/6
                           = 1/3 + 1/6 + 1/2
                           = 1/6 + 1/2 + 1/3
                           = 1/6 + 1/3 + 1/2.

Cf. A002966, A002967, A006585.

a(n) = n! * A006585(n).

cp's OEIS Frontend

A357765 Smallest positive integer that can be represented as the sum of n of its (possibly equal) divisors in the maximum number of ways (=A002966(n)).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A144063 Egyptian fractions: number of solutions of 1 = 1/x_1 + ... + 1/x_n in positive integers x_1 < ... < x_n <= 256.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A379452 Number of compositions (ordered partitions) of 1 into n distinct reciprocals of positive integers.

Views

Author

Keywords

Examples

Crossrefs

Formula