A275121 a(n) is the smallest multiple of n that is a practical number.

1, 2, 6, 4, 20, 6, 28, 8, 18, 20, 66, 12, 78, 28, 30, 16, 204, 18, 228, 20, 42, 66, 276, 24, 100, 78, 54, 28, 348, 30, 496, 32, 66, 204, 140, 36, 666, 228, 78, 40, 820, 42, 860, 88, 90, 276, 1128, 48, 196, 100, 204, 104, 1272, 54, 220, 56, 228, 348, 1416, 60

Offset

1,2

Comments

A rational in (0,1) as a fraction in lowest terms with denominator n, if expressed with denominator a(n) will have a practical-number denominator and can be written as an Egyptian fraction.

Note that a(n) exists for each n; a trivial upper bound is n * gpf(n)# = n * A034386(A006530(n)). - Charles R Greathouse IV, Jul 25 2016

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = n * A210445(n).

Example

For example a(5)=20, indicating that a fraction with denominator 5 can be rewritten as a fraction with denominator 20, which is a practical number. Thus a fraction such as 4/5 can be written as 16/20. The new numerator 16 can be written as the sum of distinct divisors of 20 (16=10+5+1) because 20 is a practical number. The fractions 10/20, 5/20, and 1/20 are each a reciprocal: 1/2, 1/4, and 1/20. Thus 4/5 can be written as the sum of distinct reciprocals (Egyptian fraction expansion) as 4/5 = 1/2 + 1/4 + 1/20.

Prog

(PARI) /* First declare the function is_a005153(n) as in A005153 */
a(n) = my(k=1); while(!is_a005153(k*n), k++); k*n \\ Felix Fröhlich, Jul 18 2016

Crossrefs

Cf. A005153 (practical numbers), A210445.

Cf. A006530, A034386.

Sequence in context: A063427 A066092 A100695 * A174824 A009262 A127699

Adjacent sequences: A275118 A275119 A275120 * A275122 A275123 A275124

Keyword

nonn

Author

Lee A. Newberg, Jul 18 2016

Extensions

More terms from Felix Fröhlich, Jul 18 2016

Status

approved