A275339 -id:A275339 - 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])

A379098 The waterproof hulls of numbers that are not waterproof (A379097).

Original entry on oeis.org

180, 252, 360, 396, 468, 504, 612, 684, 720, 792, 828, 1400, 900, 936, 1575, 1008, 1044, 1116, 1224, 6300, 2200, 1332, 1368, 1440, 1476, 2475, 1548, 2600, 4752, 1656, 2800, 1692, 2925, 1764, 1800, 4312, 5616, 3150, 1908, 9900, 2016, 3400, 4851, 2088, 2124, 5096

Offset: 1

Peter Luschny, Dec 16 2024

nonn

The waterproof hull h(k) of k is the smallest waterproof number that k divides. 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.)

Cf. A275339, A379095, A379096, A379097.

# 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) for n in range(1, 700) if WaterCapacity(n) > 0])

A331507 Water retention for an n X n number square with the maximum number of ponds using a simple filling pattern.

Original entry on oeis.org

0, 0, 1, 5, 30, 74, 195, 363, 700, 1124, 1845

Offset: 1

Craig Knecht, Jan 18 2020

The number placement starts with the lowest available number and proceeds from top left to bottom right in two separate passes. The first pass fills in the ponds. The second pass fills in the barrier cells surrounding the ponds.
A number square contains each of the numbers 1 to n*n exactly once.
The water retention model provides the definition of a pond. All the ponds have an area of 1 cell in the maximum pond example.
The immediate environment of a 1-cell pond requires four larger surrounding cells. The water retention model requires the macro environment of possible surrounding cells to be lower than the border of the 1-cell-area pond.
For even-ordered squares one of the main diagonals is made up of ponds. For odd-ordered squares both diagonals are made up of ponds.
The cells in a given row hold identical amounts of water.
A listing of the C code that calculates the water retention is given. The program gives a graphic output where the area of the ponds is color coded. Additional 3D graphics and other water retention utilites are available on Harry White's web page noted below.
The water retention model functions in three dimensions as noted in the crossrefs. The physical interpretation in three dimensions is not straightforward and the term "incarceration" of numbers is introduced.

			Order 5 square retaining 30 units of water. The positions of the remaining numbers that do not occupy the pond cells or their immediate borders are irrelevant and a zero is placed in these positions.
   0  6  0  7  0
   8  1  9  2 10
   0 11  3 12  0
  13  4 14  5 15
   0 16  0 17  0

Craig Knecht, Center of mass for the retained water
Craig Knecht, Center symmetric ponds - alternate example
Craig Knecht, Equal number of spillways and ponds
Craig Knecht, Example for the sequence
Craig Knecht, Magic squares with the maximum number of ponds
Craig L. Knecht, Walter Trump, Daniel ben-Avraham, Robert M. Ziff, Retention capacity of random surfaces, arXiv:1110.6166 [cond-mat.dis-nn], 31 Jan 2012.
Harry White, Water retention calculation and graphic display utility
Harry White, Water retention utilites.
Wikipedia, Water retention on mathematical surfaces

Cf. A201126 (water retention on magic squares), A261347 (water retention on number squares), A275359 (3 dimensional incarceration), A275339.

A379093 Numbers whose factors in the canonical prime factorization neither increase weakly nor decrease weakly but are waterproof. (Terms of A379094 but not of A379097.)

Original entry on oeis.org

90, 126, 180, 252, 270, 350, 360, 378, 504, 525, 540, 550, 594, 650, 700, 702, 756, 810, 825, 850, 918, 950, 975, 1026, 1050, 1078, 1080, 1100, 1134, 1150, 1188, 1242, 1274, 1275, 1300, 1350, 1400, 1404, 1425, 1512, 1575, 1617, 1620, 1650, 1666, 1700, 1725

Offset: 1

Peter Luschny, Dec 17 2024

nonn

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

Cf. A379094, A379095, A379096, A379097.

# Using functions 'isA379094' and 'is_not_waterproof' (from A379097).
A := select(isA379094, {seq(1..1800)}):
B := select(is_not_waterproof, {seq(1..1800)}):
A minus B;

nn = 1725; 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 *)

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

A379098 The waterproof hulls of numbers that are not waterproof (A379097).

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Python

A331507 Water retention for an n X n number square with the maximum number of ponds using a simple filling pattern.

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A379093 Numbers whose factors in the canonical prime factorization neither increase weakly nor decrease weakly but are waterproof. (Terms of A379094 but not of A379097.)

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Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica