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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. 4
 1, 2, 6, 15, 42, 143, 399, 1190, 4209, 13130, 41591, 118590, 404471, 1158696, 3893831, 12222320, 39428991, 123471920, 397952081, 1297210320 (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. The corresponding rectangle tiling provides the optimal power rating of the 1 ohm resistors with respect to the specific voltage V_s and current A_s. See the picture From_Quilt_to_Net in the link section, which also provides insight in the "mysterious" correspondence between rectangle tilings and electric networks. For non-planar nets the idea of rectangle tilings can be widened to 'Cartesian squarings'. A Cartesian squaring is the dissection of the product P X Q of two finite sets into 'squaresets', i.e., sets A X B with A subset of P and B subset of Q, and card(A) = card(B). - Rainer Rosenthal, Dec 14 2022 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..20. Rainer Rosenthal, From_Quilt_to_Net 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 A373455 A148438 Adjacent sequences: A340723 A340724 A340725 * A340727 A340728 A340729 KEYWORD nonn,hard,more,nice 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 a(19)-a(20) from Hugo Pfoertner, Apr 16 2021 STATUS approved

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Last modified September 18 01:28 EDT 2024. Contains 375995 sequences. (Running on oeis4.)