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A340726 Maximum power V_s*A_s consumed by an electrical network with n unit resistors and input voltage V_s and current A_s constrained to be exact integers which are coprime, and such that all currents between nodes are integers. 1
1, 2, 6, 15, 42, 143, 399, 1190, 4209, 13130, 41591, 118590, 404471, 1158696, 3893831, 12222320, 39428991, 123471920 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

This sequence is an analog of A338861. Equality a(n) = A338861(n) holds for small n only, see example.

Let V_s denote the specific voltage, i.e., the lowest integer voltage, which induces integer currents everywhere in the network. Denote by A_s the specific current, i.e., the corresponding total current.

A planar network with n unit resistors corresponds to a squared rectangle with height V_s and width A_s. The electrical power V_s*A_s therefore equals the area of that rectangle. In the historical overview (Stuart Anderson link) A_s is called complexity.

Take the set SetA337517(n) of resistances, counted by A337517. For each resistance R multiply numerator and denominator. Conjecture: a(n) is the maximum of all these products. The reason is that common factors of V_s and A_s are quite rare (see the beautiful exceptional example with 21 resistors).

LINKS

Table of n, a(n) for n=1..18.

Squaring.Net 2020, Stuart Anderson, Squared Rectangle and Smith Diagram

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.

n = 21:

The electrical network corresponding to the perfect squared square A014530 has specific voltage V_s equal to specific current A_s, namely V_s = A_s = 112. Its power V_s*A_s = 12544 is far below the maximum a(20) > a(10) > 13000, and a(n) is certainly monotonically increasing. - Rainer Rosenthal, Mar 28 2021

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,changed

AUTHOR

Rainer Rosenthal, Jan 17 2021

EXTENSIONS

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

Definition corrected by Rainer Rosenthal, Mar 28 2021

a(18) from Hugo Pfoertner, Apr 09 2021

STATUS

approved

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Last modified April 12 18:58 EDT 2021. Contains 342932 sequences. (Running on oeis4.)