A379096 Waterproof numbers >= 60.

61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121

Offset: 1

Author

Peter Luschny, Dec 16 2024

Keywords

• nonn

Comments

All nonnegative numbers less than 60 are waterproof.

Zero and one are waterproof numbers by convention. Numbers that admit a prime factorization are waterproof if their water capacity is 0. (The water capacity of a number is defined in A275339.)

If the factors p_i^e_i in the canonical prime factorization of n are weakly ascending or weakly descending, then n is waterproof.

A number is waterproof if and only if it equals its waterproof hull (A379098). The waterproof hull h(n) of n is the smallest waterproof number that n divides.

Numbers that are not waterproof are listed in A379097.

Examples

			Numbers having at most two distinct prime factors (A070915) are waterproof. The primorials (A002110) are waterproof.
48300 has a water capacity of 17 and so is not waterproof. The waterproof hull of 48300 is 1014300.
		

Crossrefs

Cf. A275339, A379094, A379095, A379097, A379098.

Programs

Maple

# The function 'water_capacity' is defined in A275339.
is_waterproof := n -> ifelse(n < 2, true, is(water_capacity(n) = 0)):
select(is_waterproof, [seq(60..121)]);

Python

# The function 'WaterCapacity' is defined in A275339.
print([n for n in range(60, 122) if WaterCapacity(n) == 0])