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

Author

Lee A. Newberg, Jul 18 2016

Keywords

• nonn

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A005153 (practical numbers), A210445.

Cf. A006530, A034386.

Programs

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

Formula

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

Extensions

More terms from Felix Fröhlich, Jul 18 2016