A385811 Numbers k such that there exists a partition of the sum of prime factors of k (cf. A001414) into bigomega(k) (cf. A001222) prime parts where the product of parts is more than k.

20, 21, 28, 33, 39, 40, 42, 44, 51, 52, 56, 57, 60, 63, 65, 66, 68, 69, 76, 78, 80, 84, 85, 87, 88, 92, 93, 95, 99, 100, 102, 104, 105, 111, 112, 114, 115, 116, 117, 119, 120, 123, 124, 126, 129, 130, 132, 133, 136, 138, 140, 141, 145, 147, 148, 152, 153, 155

Offset: 1

Author

Gordon Hamilton, Jul 09 2025

Keywords

• nonn

Comments

As a temporary name, let's call these numbers "inefficient".

A number n is inefficient if there is a larger number N which has the same number of prime factors (counted with multiplicity) and the sum of the prime factors of n and N are the same.

The density of terms in the positive integers is 1.

This is a good sequence for students exploring prime factorization for the first time. When teaching elementary school students, refrain from divulging the rules upfront. Instead, seek high engagement by delivering a series of epic communal fails. I ask the students to give me an inefficient number, knowing they know nothing about what that means. When they supply "15" I'll show that 3*5 = 15 and 3+5 = 8 and complain that they've failed. When they do stumble upon a number like "28" then I use the opportunity to explain more of the rules.

Some numbers are noticeably absent from this list:

- primes,

- powers of primes,

- numbers whose prime factors include only two consecutive prime numbers,

- double any of the above three.

Examples

			60 is inefficient because its prime factors are 2,2,3,5. These factors add to 12. 81 is larger than 60 and also has its four prime factors adding to 12. 60 is therefore inefficient.
63 is inefficient because its prime factors are 3,3,7. These factors add to 13. 75 is larger than 63 and also has its three prime factors (3,5,5) adding to 13. 63 is therefore inefficient.
		

Crossrefs

Cf. A001222, A001414, A385755, A385756.

Extensions

More terms from Alois P. Heinz, Jul 09 2025