A338602 -id:A338602 - cp's OEIS frontend

A338601 Numerators x of resistance values R=x/y that can be obtained by a network of at most 10 one-ohm resistors such that a network of more than 10 one-ohm resistors is needed to obtain the resistance y/x. Denominators are in A338602.

95, 101, 98, 97, 103, 97, 110, 103, 130, 103, 115, 106, 109, 98, 101

Offset: 1

Hugo Pfoertner, Nov 08 2020

The terms are sorted by increasing value of the resistance R(n) = a(n)/A338602(n).
For more information, references, and links, see A180414 and A338573.
Each network for R = p/q is visualized (see link section) as a multigraph with the battery nodes on top and at the bottom, i.e., the battery edge does not have to cross any edges. Any planar network with 10 resistors presented in this way, has a corresponding tiling of a p X q rectangle by 10 squares, and the inverse resistance q/p can be obtained in the same way. According to the definition of a(n) this is not the case here, so there must be crossing edges in every drawing. It should be noticed though, that all the networks (without the battery edge) are planar. - Rainer Rosenthal, Jan 03 2021
Version 2 of the visualization (see link section) shows that all these exceptional networks are extensions of the same network with 8 resistors. It is the graph K_3_3 without the 'battery edge' A-Z and shall be named VG8. This network VG8 has no related squared rectangle, because it has no series-parallel subnets and has resistance 5/4, but there is no such network with resistance 4/5. So, this is the graph, which is mentioned by Karnofsky in his "Addendum": "The smallest non-planar graph has eight resistors.". - Rainer Rosenthal, Feb 13 2021
The reciprocal 101/130 of R(9) = 130/101 needs 12 resistors, while the other 14 reciprocal resistance values can be obtained by networks of 11 resistors. - Rainer Rosenthal, Jan 16 2021

			All fractions for 10 resistors are: 95/106, 101/109, 98/103, 97/98, 103/101, 97/86, 110/91, 103/83, 130/101, 103/80, 115/89, 106/77, 109/77, 98/67, 101/67.
The corresponding networks are shown below, with -(always 1) and +(maximum node number) indicating the nodes where the voltage is applied. Edges marked ==, ||, //, or \\, have 2 resistors in parallel.
.
     95/106        101/109         98/103         97/98         103/101
  -1=======2     -1-------2     -1-------2     -1-------2     -1-------2
   |\     /|      |\     /||     |\     /|      |\     /|      |\     /|
   | \   / |      | \   / ||     | \   / |      | \   / |      | \   / |
   |  \ /  |      |  \ /  ||     |  \ /  |      |  \ /  |      |  \ /  |
   |   4   |      |   4   ||     |   4   |      |   6   4      4   6   |
   |  / \  |      |  / \  ||     |  //\  |      |  / \  |      |  / \  |
   | /  +6 |      | /  +6 ||     | // +6 |      | /  +7 |      | /  +7 |
   |/     \|      |/     \||     |//    \|      |/     \|      |/     \|
   3-------5      3-------5      3-------5      3-------5      3-------5
.
     97/86         110/91         103/83         130/101        103/80
  -1=======2         -1         -1-------2     -1-----2       -1=======2
   |      /|         / \         |      /||     |    /|\       |      /|
   |     / |        /   \        |     4 ||     |   | | |      |     4 |
   |    /  |       2-----3       |    /  ||     |   | | |      |    /| |
   |   6   |      ||\   / \      |   6   ||     |   4-6 |      |   / 6 |
   |  / \  |      || \4/  |      |  / \  ||     |  /  | |      |  /  | |
   | 4  +7 |      ||  \   |      | /  +7 ||     | /  +7 |      | /  +7 |
   |/     \|       \\ +6--5      |/     \||     |/     \|      |/     \|
   3-------5        \\===//      3-------5      3-------5      3-------5
.
    115/89         106/77         109/77          98/67         101/67
     -1          -1-------2     -1-------2     -1-------2     -1-------2
     / \          |      /||     |     //|      |      /|      |      /|
    /   \         |     4 ||     |     4 |      |     6 |      |     6 |
   2-----3        |    /| ||     |    /| |      |    /| |      |    /| |
   |\   / \       |   / 6 ||     |   / 6 |      |   / 7 |      |   / 7 4
   | \6/  |       |  /  | ||     |  /  | |      |  4  | |      |  /  | |
   |  \   |       | /  +7 ||     | /  +7 |      | /  +8 |      | /  +8 |
   |  +7--5       |/     \||     |/     \|      |/     \|      |/     \|
   4------/       3-------5      3-------5      3-------5      3-------5

Allan Gottlieb, Oct 3, 2003 addendum (Karnofsky).
Rainer Rosenthal, Karnofsky's 15 exceptional networks with 10 resistors (Version 2).
Index to sequences related to resistances.

Cf. A180414, A338197, A338573, A338580, A338590, A338602.

Cf. A338581, A338591, A338582, A338592 (similar for n = 11 and n = 12).

A338590 Denominators of resistance values that can be obtained from a network of exactly 10 one-ohm resistors, but not from any network with fewer than 10 one-ohm resistors. Numerators are in A338580.

Original entry on oeis.org

10, 17, 23, 22, 27, 33, 32, 25, 31, 37, 41, 35, 29, 40, 45, 39, 38, 43, 37, 57, 46, 51, 54, 49, 63, 43, 52, 47, 51, 60, 55, 41, 50, 59, 58, 49, 40, 53, 57, 48, 61, 43, 47, 64, 38, 59, 67, 71, 29, 33, 78, 45, 49, 53, 57, 61, 73, 75, 63, 59, 55, 51, 47, 82, 35, 31

Offset: 1

Hugo Pfoertner, Nov 06 2020

			The list of the 2639 = A338197(10) resistance values, sorted by increasing size of R = A338580(n)/a(n), is [1/10, 2/17, 3/23, 3/22, 4/27, 5/33, 5/32, ..., 32/5, 33/5, 27/4, 22/3, 23/3, 17/2, 10]. There are 15 terms for which their reciprocal value is not in the sequence, given in A338601/A338602.

Hugo Pfoertner, Table of n, a(n) for n = 1..2639

Cf. A180414, A338197, A338573, A338580, A338600, A338601, A338602.

Cf. A338595, A338596, A338597, A338599, A338599 (similar for n = 5..10).

Title corrected by Rainer Rosenthal, Feb 14 2021

A338573 Array read by ascending antidiagonals: T(m,n) (m, n >= 1) is the minimum number of unit resistors needed to produce resistance m/n.

Original entry on oeis.org

1, 2, 2, 3, 1, 3, 4, 3, 3, 4, 5, 2, 1, 2, 5, 6, 4, 4, 4, 4, 6, 7, 3, 4, 1, 4, 3, 7, 8, 5, 2, 5, 5, 2, 5, 8, 9, 4, 5, 3, 1, 3, 5, 4, 9, 10, 6, 5, 5, 5, 5, 5, 5, 6, 10, 11, 5, 3, 2, 5, 1, 5, 2, 3, 5, 11, 12, 7, 6, 6, 5, 5, 5, 5, 6, 6, 7, 12, 13, 6, 6, 4, 6, 4, 1, 4, 6, 4, 6, 6, 13

Offset: 1

Rainer Rosenthal, Nov 05 2020

Karnofsky (2004, p. 5): "[...] if some circuit has resistance m/n then some other circuit likely has n/m. In fact, for 9 or fewer resistors, this symmetry is perfect. However, for 10 resistors the following values are achieved, but not their inverses: 95/106, 101/109, 98/103, 97/98, 103/101, 97/86, 110/91, 103/83, 130/101, 103/80, 115/89, 106/77, 109/77, 98/67, 101/67". That means, that T(m,n) = T(n,m), if T(m,n) <= 9.
This starts with the values of A113881, but the Karnofsky comment says that T(n,m) is not symmetric, whereas the count of tiles in A113881 is. - R. J. Mathar, Nov 06 2020
The first difference where T(m,n) = T(n,m), but differs from the corresponding entry of A113881 occurs for (n,m) = (154,167) and (n,m) = (167,154), both representable by networks with non-planar graphs of 11 resistors, whereas A113881 counts 12 tiles. See Pfoertner link for illustration of more differences. - Hugo Pfoertner, Nov 13 2020

			T(1,2) = 2: at least 2 unit resistors in parallel are needed for resistance 1/2.
T(2,1) = 2: at least 2 unit resistors in series are needed for resistance 2 = 2/1.
T(11,13) = 6: the following "bridge" has resistance Bri(Par(1,1),1,1,1,1) = 11/13 (see A337516 for definitions):
.
                  (+)
                  / \
              ---*   \
             /  /     \
           (1)(1)     (1)
             \ |       |
              \|       |
               *--(1)--*
                \     /
                (1) (1)
                  \ /
                  (-)
.
T(13,11) = 6: Bri(Ser(1,1),1,1,1,1) = 13/11.
T(95,106) = 10, but T(106,95) > 10: Karnofsky (2004, p. 5), see comment.

Technology Review's Puzzle Corner, How many different resistances can be obtained by combining 10 one ohm resistors? Oct 3, 2003.

Joel Karnofsky, Solution of problem from Technology Review's Puzzle Corner Oct 3, 2003, Feb 23 2004.
Hugo Pfoertner, Where A338573 differs from A113881, x,y <= 380.
Index to sequences related to resistances.

Cf. A048211, A113881, A180414, A174283, A337516, A337517.

Non-reciprocal ratios: A338601/A338602 (10 resistors), A338581/A338591 (11 resistors), A338582/A338592 (12 resistors).

A338581 Numerators x of resistance values R=x/y that can be obtained by a network of at most 11 one-ohm resistors such that a network of more than 11 one-ohm resistors is needed to obtain the resistance y/x. Denominators are in A338591.

Original entry on oeis.org

95, 101, 98, 97, 103, 97, 110, 103, 130, 103, 115, 106, 109, 119, 106, 98, 116, 109, 101, 124, 121, 111, 121, 136, 124, 151, 141, 169, 121, 151, 134, 136, 125, 133, 127, 134, 136, 149, 142, 146, 161, 137, 161, 142, 146, 145, 152, 149, 169, 161, 151, 167, 175, 149, 151, 194, 176, 150, 166, 174

Offset: 1

Hugo Pfoertner, Nov 08 2020

The terms are sorted by increasing value of R.

See A338601 for more information.

Hugo Pfoertner, Table of n, a(n) for n = 1..172
Hugo Pfoertner, Visualization of the non-reciprocal resistance values with 11 resistors, using Plot 2.

Cf. A180414, A338591, A338592, A338601, A338602.

First differs from A338582 for n=59.

A338591 Denominators y of resistance values R=x/y that can be obtained by a network of at most 11 one-ohm resistors such that a network of more than 11 one-ohm resistors is needed to obtain the resistance y/x. Numerators are in A338581.

Original entry on oeis.org

201, 210, 201, 195, 204, 183, 201, 186, 231, 183, 204, 183, 186, 201, 179, 165, 195, 183, 168, 191, 186, 169, 183, 199, 175, 209, 194, 231, 164, 204, 181, 183, 168, 178, 169, 173, 175, 191, 179, 181, 199, 167, 194, 169, 173, 166, 167, 162, 183, 174, 161, 178, 186

Offset: 1

Hugo Pfoertner, Nov 08 2020

Hugo Pfoertner, Table of n, a(n) for n = 1..172

Cf. A180414, A338591, A338601, A338602.

A338582 Numerators x of resistance values R=x/y that can be obtained by a network of at most 12 one-ohm resistors such that a network of more than 12 one-ohm resistors is needed to obtain the resistance y/x. Denominators are in A338592.

Original entry on oeis.org

95, 101, 98, 97, 103, 97, 110, 103, 130, 103, 115, 106, 109, 119, 106, 98, 116, 109, 101, 124, 121, 111, 121, 136, 124, 151, 141, 169, 121, 151, 134, 136, 125, 133, 127, 134, 136, 149, 142, 146, 161, 137, 161, 142, 146, 145, 152, 149, 169, 161, 151, 167, 175, 149, 151, 194, 176, 150, 152, 166

Offset: 1

Hugo Pfoertner, Nov 08 2020

The terms are sorted by increasing value of R.

See A338601 for more information.

Hugo Pfoertner, Table of n, a(n) for n = 1..1114
Hugo Pfoertner, Visualization of the non-reciprocal resistance values with 12 resistors, using Plot 2.
Hugo Pfoertner, Comparison of numerators for 11 and 12 resistors, using Plot 2.

Cf. A180414, A338581, A338591, A338592, A338601, A338602.

A338592 Denominators y of resistance values R=x/y that can be obtained by a network of at most 12 one-ohm resistors such that a network of more than 12 one-ohm resistors is needed to obtain the resistance y/x. Numerators are in A338582.

Original entry on oeis.org

296, 311, 299, 292, 307, 280, 311, 289, 361, 286, 319, 289, 295, 320, 285, 263, 311, 292, 269, 315, 307, 280, 304, 335, 299, 360, 335, 400, 285, 355, 315, 319, 293, 311, 296, 307, 311, 340, 321, 327, 360, 304, 355, 311, 319, 311, 319, 311, 352, 335, 312, 345, 361

Offset: 1

Hugo Pfoertner, Nov 08 2020

Hugo Pfoertner, Table of n, a(n) for n = 1..1114

Cf. A180414, A338581, A338582, A338591, A338601, A338602.

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.

Original entry on oeis.org

1, 2, 6, 15, 42, 143, 399, 1190, 4209, 13130, 41591, 118590, 404471, 1158696, 3893831, 12222320, 39428991, 123471920, 397952081, 1297210320

Offset: 1

Rainer Rosenthal, Jan 17 2021

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).

			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

Rainer Rosenthal, From_Quilt_to_Net
Squaring.Net 2020, Stuart Anderson, Squared Rectangle and Smith Diagram
Index to sequences related to resistances.

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

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

A340920 a(n) is the number of distinct resistances that can be produced from a planar circuit with exactly n unit resistors.

Original entry on oeis.org

1, 1, 2, 4, 9, 23, 57, 151, 427, 1263, 3807, 11549, 34843, 104459, 311317, 928719, 2776247, 8320757, 24967341, 74985337

Offset: 0

Hugo Pfoertner and Rainer Rosenthal, Feb 14 2021

			a(10) = 3807, whereas A337517(10) = 3823. The difference of 16 resistances results from the 15 terms of A338601/A338602 and the resistance 34/27 not representable by a planar network of 10 resistors, whereas it (but not 27/34) can be represented by a nonplanar network of 10 resistors.

Stuart Anderson, Catalogues of Simple Perfect Squared Rectangles (SPSR).
Stuart Anderson, Simple Imperfect Squared Rectangles, orders 9 to 24.

Cf. A002840, A048211, A174283, A180414, A337516, A337517, A338511, A338601, A338602, A340921.

Cf. A113881, A219158.

Replace the list of 3-connected graphs corresponding to A338511 by the list of planar graphs provided in A002840 in Andrew Howroyd's PARI program linked in A180414.

a(n) = A337517(n) for n <= 9, a(n) < A337517(n) for n >= 10.

a(19) from Hugo Pfoertner, Mar 15 2021

A339547 a(n) is the number of resistance values R=x/y that can be obtained by a network of at most n one-ohm resistors such that a network of more than n one-ohm resistors is needed to obtain the resistance y/x.

Original entry on oeis.org

15, 172, 1114, 5378, 22321, 83995, 293744, 968965

Offset: 10

Hugo Pfoertner, Dec 10 2020

a(n) = 0 for n < 10.

			a(10) = 15: this is the number of non-reciprocal resistance values provided in Karnofsky's solution of the 10-resistors puzzle. The list of 15 resistances is: 95/106, 101/109, 98/103, 97/98, 103/101, 97/86, 110/91, 103/83, 130/101, 103/80, 115/89, 106/77, 109/77, 98/67, 101/67.
a(11) = 172: the corresponding resistances are provided in A338581/A338591.
a(12) = 1114: the corresponding resistances are provided in A338582/A338592.

Technology Review's Puzzle Corner, How many different resistances can be obtained by combining 10 one ohm resistors? Oct 3, 2003.

Joel Karnofsky, Solution of problem from Technology Review's Puzzle Corner Oct 3, 2003, Feb 23 2004.

Cf. A180414, A338573, A338601, A338602, A338581, A338591, A338582, A338592.

cp's OEIS Frontend

A338601 Numerators x of resistance values R=x/y that can be obtained by a network of at most 10 one-ohm resistors such that a network of more than 10 one-ohm resistors is needed to obtain the resistance y/x. Denominators are in A338602.

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A338590 Denominators of resistance values that can be obtained from a network of exactly 10 one-ohm resistors, but not from any network with fewer than 10 one-ohm resistors. Numerators are in A338580.

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A338573 Array read by ascending antidiagonals: T(m,n) (m, n >= 1) is the minimum number of unit resistors needed to produce resistance m/n.

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A338581 Numerators x of resistance values R=x/y that can be obtained by a network of at most 11 one-ohm resistors such that a network of more than 11 one-ohm resistors is needed to obtain the resistance y/x. Denominators are in A338591.

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A338591 Denominators y of resistance values R=x/y that can be obtained by a network of at most 11 one-ohm resistors such that a network of more than 11 one-ohm resistors is needed to obtain the resistance y/x. Numerators are in A338581.

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A338582 Numerators x of resistance values R=x/y that can be obtained by a network of at most 12 one-ohm resistors such that a network of more than 12 one-ohm resistors is needed to obtain the resistance y/x. Denominators are in A338592.

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A338592 Denominators y of resistance values R=x/y that can be obtained by a network of at most 12 one-ohm resistors such that a network of more than 12 one-ohm resistors is needed to obtain the resistance y/x. Numerators are in A338582.

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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.

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A340920 a(n) is the number of distinct resistances that can be produced from a planar circuit with exactly n unit resistors.

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A339547 a(n) is the number of resistance values R=x/y that can be obtained by a network of at most n one-ohm resistors such that a network of more than n one-ohm resistors is needed to obtain the resistance y/x.

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