cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A379097 Numbers that are not waterproof.

Original entry on oeis.org

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

Views

Author

Peter Luschny, Dec 16 2024

Keywords

Comments

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

Links

Crossrefs

Cf. A275339, A379094, A379096, A379098.

Programs

  • Maple
    # 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)]);
  • Mathematica
    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 *)
  • Python
    # The function 'WaterCapacity' is defined in A275339.
print([n for n in range(818) 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

Views

Author

Peter Luschny, Dec 16 2024

Keywords

Comments

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.

Examples

			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).

Crossrefs

Cf. A275339, A379096, A379097, A379098.

Programs

  • Python
    # 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

Views

Author

Peter Luschny, Dec 16 2024

Keywords

Comments

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.)

Crossrefs

Cf. A275339, A379095, A379096, A379097.

Programs

  • Python
    # 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])

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

Views

Author

Peter Luschny, Dec 17 2024

Keywords

Links

Crossrefs

Cf. A379094, A379095, A379096, A379097.

Programs

  • Maple
    # 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;
  • Mathematica
    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 *)
Showing 1-4 of 4 results.

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