A379098 -id:A379098 - cp's OEIS frontend

A379097 Numbers that are not waterproof.

60, 84, 120, 132, 156, 168, 204, 228, 240, 264, 276, 280, 300, 312, 315, 336, 348, 372, 408, 420, 440, 444, 456, 480, 492, 495, 516, 520, 528, 552, 560, 564, 585, 588, 600, 616, 624, 630, 636, 660, 672, 680, 693, 696, 708, 728, 732, 744, 760, 765, 780, 804, 816

Offset: 1

Peter Luschny, Dec 16 2024

nonn

Zero and one are waterproof numbers by convention. Numbers that admit a prime factorization are not waterproof if their water capacity is > 0. (The water capacity of a number is defined in A275339.)

Proper subset of A375055, in turn a proper subset of A126706, since A001221(a(n)) >= 3 and a maximum multiplicity is required for at least one prime power factor, so as to have positive water capacity. - Michael De Vlieger, Dec 18 2024

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A275339, A379094, A379096, A379098.

# The function 'water_capacity' is defined in A275339.
is_not_waterproof := n -> ifelse(n < 2, false, is(water_capacity(n) <> 0)):
select(is_not_waterproof, [seq(0..820)]);

nn = 816;
s = Select[Range[nn], Nor[SquareFreeQ[#], PrimePowerQ[#]] &];
Select[s, Function[f, And[NoneTrue[{Sort[f], ReverseSort[f]}, # == f &],
  Total[(f //. {a___, b_, c__, d_, e___} /;
    AllTrue[{c}, And[# < b, # < d] &] :>
    {a, b, Sequence @@ Table[Min[b, d], {Length[{c}]}], d, e}) - f] > 0] ]
[Power @@@ FactorInteger[#]] &] (* Michael De Vlieger, Dec 18 2024, after Jean-François Alcover at A275339 *)

# The function 'WaterCapacity' is defined in A275339.
print([n for n in range(818) if WaterCapacity(n) > 0])

A379096 Waterproof numbers >= 60.

Original entry on oeis.org

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

Peter Luschny, Dec 16 2024

nonn

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.

			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.

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

# 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)]);

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

A379095 The water sealings of numbers that are not waterproof (A379097).

Original entry on oeis.org

3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 3, 3, 5, 3, 3, 3, 3, 15, 5, 3, 3, 3, 3, 5, 3, 5, 9, 3, 5, 3, 5, 3, 3, 7, 9, 5, 3, 15, 3, 5, 7, 3, 3, 7, 3, 3, 5, 5, 15, 3, 9, 7, 15, 3, 5, 3, 5, 3, 9, 5, 21, 5, 3, 7, 3, 3, 5, 3, 15, 3, 5, 5, 9, 7, 3, 7, 21, 9, 5, 3, 15, 5

Offset: 1

Peter Luschny, Dec 16 2024

nonn

The water sealing of a number n is the smallest positive integer s(n) so that the water hull of n can be written h(n) = n * s(n). n is waterproof if and only if s(n) = 1.

			48300 has a water capacity of 17 and so is not waterproof. The waterproof hull of 48300 is 1014300. Thus the sealing of 48300 is 21. The prime factorization of the sealing shows where the water holes of n are, in this example at 3 and 7 (see the example in A275339).

Cf. A275339, A379096, A379097, A379098.

# Using function "WaterCapacity" from A275339.
def s(n: int) -> int:
    j = n
    while True:
        if WaterCapacity(j) == 0 and j % n == 0: return j
        j += n
print([s(n)//n for n in range(1, 1200) if WaterCapacity(n) > 0])

cp's OEIS Frontend

A379097 Numbers that are not waterproof.

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A379096 Waterproof numbers >= 60.

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A379095 The water sealings of numbers that are not waterproof (A379097).

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Python