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A340726 Maximum power V*I consumed by an electrical network with n unit resistors and input voltage V and current I constrained to be exact integers which are coprime. 0
1, 2, 6, 15, 42, 143, 399, 1190, 4209, 13130, 41591, 118590, 404471, 1158696, 3893831, 12222320, 39428991 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Take the set SetA337517(n) of resistances, counted by A337517. For each resistance R multiply numerator and denominator. Then a(n) is the maximum of all these products.

This sequence is an analog of A338861.

Any planar network with n unit resistors with resistance R = V/I corresponds to a rectangle with same ratio R = height/width, which is tiled by n squares. The electrical power V*I therefore corresponds to the area height*width of the corresponding tiled rectangle.

Equality a(n) = A338861(n) holds for small n only, see example.

LINKS

Table of n, a(n) for n=1..17.

Index to sequences related to resistances.

EXAMPLE

n = 3:

Networks with 3 unit resistors have A337517(3) = 4 resistance values: {1/3, 3, 3/2, 2/3}. The maximum product numerator x denominator is 6.

n = 6:

Networks with 6 unit resistors have A337517(6) = 57 resistance values, where 11/13 and 13/11 are the resistances with maximum product numerator x denominator.

                                             +-----------+-------------+

                     A                       |           |             |

                    / \                      |           |             |

               (1) /   \ (2)                 |   6 X 6   |    7 X 7    |

                  /     \                    |           |             |

                 /  (3)  \                   |           |             |

                o---------o                  +---------+-+             |

                 \       //                  |         +-+-----+-------+

                  \  (5)//                   |  5 X 5  |       |       |

               (4) \   //(6)                 |         | 4 X 4 | 4 X 4 |

                    \ //                     |         |       |       |

                     Z                       +---------+-------+-------+

       ___________________________________________________________________

        Network with 6 unit resistors       Corresponding rectangle tiling

        total resistance 11/13 giving          with 6 squares giving

            a(6) = 11 x 13 = 143                 A338861(6) = 143

n = 10:

With n = 10, non-planarity comes in, yielding a(10) > A338861(10).

The "culprit" here is the network with resistance A338601(9)/A338602(9) = 130/101, giving a(10) = 13130 > A338861(10) = 10920.

CROSSREFS

Cf. A180414, A337517, A338601, A338602, A338861.

Sequence in context: A178936 A221744 A338861 * A303833 A148438 A148439

Adjacent sequences:  A340723 A340724 A340725 * A340727 A340728 A340729

KEYWORD

nonn,hard,more

AUTHOR

Rainer Rosenthal, Jan 17 2021

EXTENSIONS

a(13)-a(17) from Hugo Pfoertner, Feb 08 2021

STATUS

approved

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Last modified February 25 22:37 EST 2021. Contains 341618 sequences. (Running on oeis4.)