A338197 -id:A338197 - cp's OEIS frontend

A338600 a(n) is the common denominator of the A338197(n) rational resistance values that can be obtained from a network of exactly n one-ohm resistors, but not by a network of fewer than n one-ohm resistors.

1, 2, 6, 60, 840, 360360, 232792560, 5342931457063200, 591133442051411133755680800, 79057815923102180093748328364591874435251553600

Offset: 1

Hugo Pfoertner, Nov 03 2020

nonn

The next terms a(11)=8.87124454467...*10^84 and a(12)=1.80685581583...*10^141 are too big to be included in the data.

			a(4) = 60: The resistance values for which a minimum of 4 resistors is needed are [1/4, 2/5, 3/5, 3/4, 4/3, 5/3, 5/2, 4] with a common denominator of 60.
a(1) = 1: [1],
a(2) = 2: [1/2, 2],
a(3) = 6: [1/3, 2/3, 3/2, 3].

Hugo Pfoertner, Table of n, a(n) for n = 1..15

Cf. A048211, A051389, A180414, A337517, A338197, A338580, A338590, A338595, A338596, A338597, A338598, A338599, A338605, A338606, A338607, A338608, A338609.

A180414 Number of different resistances that can be obtained by combining n one-ohm resistors.

Original entry on oeis.org

1, 2, 4, 8, 16, 36, 80, 194, 506, 1400, 4039, 12044, 36406, 111324, 342447, 1064835, 3341434, 10583931, 33728050, 107931849, 346616201

Offset: 0

Vaclav Kotesovec, Sep 02 2010

In "addendum" J. Karnofsky stated the value a(15) = 1064833. In contrast to the terms up to and including a(14), which could all be confirmed, an independent calculation based on a list of 3-connected simple graphs resulted in the corrected value a(15) = 1064835. - Hugo Pfoertner, Dec 06 2020

See A337517 for the number of different resistances that can be obtained by combining /exactly/ n one-ohm resistors. The method used by Andrew Howroyd (see his program in the link section) uses 3-connected graphs with one edge (the 'battery edge') removed. - Rainer Rosenthal, Feb 07 2021

			a(n) counts all resistances that can be obtained with fewer than n resistors as well as with exactly n resistors. Without a resistor the resistance is infinite, i.e., a(0) = 1. One 1-ohm resistor adds resistance 1, so a(1) = 2. Two resistors in parallel give 1/2 ohm, while in series they give 2 ohms. So a(2) is the number of elements in the set {infinity, 1, 1/2, 2}, i.e., a(2) = 4. - _Rainer Rosenthal_, Feb 07 2021

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

Allan Gottlieb, Oct 3, 2003 addendum (Karnofsky).
Andrew Howroyd, PARI program (includes scripts for other related sequences)
Joel Karnofsky, Solution of problem from Technology Review's Puzzle Corner Oct 3, 2003, Feb 23 2004.
Joel Karnofsky, Mathematica program for resistances, copied from article.
Index to sequences related to resistances.

Cf. A048211, A051389, A174284, A337517, A338197, A338487, A338573, A338580, A338590, A338600.

Cf. A123545, A338511, A338999, A339072.

Mathematica
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(* See link. *)
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a(n) = A174284(n) + 1 for n <= 7, a(n) > A174284(n) + 1 otherwise. - Hugo Pfoertner, Nov 01 2020

a(n) is the number of elements in the union of the sets SetA337517(k), k <= n, counted by A337517. - Rainer Rosenthal, Feb 07 2021

a(15) corrected and a(16) added by Hugo Pfoertner, Dec 06 2020
a(17) from Hugo Pfoertner, Dec 09 2020
a(0) from Rainer Rosenthal, Feb 07 2021
a(18) from Hugo Pfoertner, Apr 09 2021
a(19) from Zhao Hui Du, May 15 2023
a(20) from Zhao Hui Du, May 23 2023

A337517 a(n) is the number of distinct resistances that can be produced from a circuit with exactly n unit resistors.

Original entry on oeis.org

1, 1, 2, 4, 9, 23, 57, 151, 427, 1263, 3823, 11724, 36048, 110953, 342079, 1064468, 3341067, 10583564, 33727683, 107931482, 346615834

Offset: 0

Rainer Rosenthal and Hugo Pfoertner, Oct 29 2020

One can view a circuit with n unit resistors as a multigraph G with n edges and a pair P of distinguished nodes. Every edge of the graph must be contained in a path connecting the two distinguished nodes.
In case n > 0, a(n) counts all resistances R(G, P), which are rational numbers by Kirchhoff's laws. In case n = 0, the graph G consists of only two pair P nodes, and there is only one resistance: oo = infinity; so a(0) = 1. In the OEIS, there are already sequences that count the possible resistances of circuits of certain types (for the definitions see A337516).
     OEIS  | type |  1  2  3  4   5   6   7    8     9    10    11     12     13
  ---------+------+--------------------------------------------------------------
   A048211 | SP   | [1] 2  4  9  22  53  131  337   869  2213  5691  14517  37017
   A174283 | SPB  |  1  2  4  9  23 [57] 151  415  1157  3191  8687  23199  61677
   A337516 | SPBF |  1  2  4  9  23  57  151 [421] 1202  3397  9498  25970  70005
   A337517 | all  |  1  2  4  9  23  57  151 [427] 1263  3823 11724  36048 110953
The table shows the number of different resistances, which grows with the complexity of the circuits. Values in square brackets mark the beginning of the newly explored range. Values a(n) up to n = 7 are fully classified, and have one of the given types, i.e., they can be computed by the functions Ser(), Par(), Bri(), and Frk() defined in A337516. For a(n), n >= 8, the theory in A180414 has to be applied.
Note: The 'set counted by A180414(n)' is the union of all 'sets counted by A337517(k) for k = 0 .. n'.
Admissible networks (G, P) are those defined in the Karnofsky paper (A180414).

			For a(n) up to n = 7 see the above mentioned sequences.

Wikipedia, Electrical resistance and conductance
Index to sequences related to resistances.

Cf. A048211, A180414, A174283, A337516, A338197.

a(8)-a(14) from Andrew Howroyd, Oct 31 2020
a(15)-a(16) from Hugo Pfoertner, Dec 06 2020
a(17) from Hugo Pfoertner, Dec 09 2020
a(18) from Hugo Pfoertner, Apr 09 2021
a(19) from Zhao Hui Du, May 15 2023
a(20) from Zhao Hui Du, May 23 2023

A338580 Numerators 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. Denominators are in A338590.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 5, 4, 5, 6, 7, 6, 5, 7, 8, 7, 7, 8, 7, 11, 9, 10, 11, 10, 13, 9, 11, 10, 11, 13, 12, 9, 11, 13, 13, 11, 9, 12, 13, 11, 14, 10, 11, 15, 9, 14, 16, 17, 7, 8, 19, 11, 12, 13, 14, 15, 18, 19, 16, 15, 14, 13, 12, 21, 9, 8, 15, 20, 19, 17, 11, 19, 14

Offset: 1

Hugo Pfoertner, Nov 06 2020

The A338197(k)/2 initial terms of this sequence are also the numerators of the corresponding sorted lists of resistance values that can be achieved by exactly k resistors for all k < 10.

			See A338590.

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

Cf. A180414, A338197, A338590.

The initial terms of this sequence are also the numerators to be used together with the denominators given in A338595, A338596, A338597, A338598, A338599.

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.

Original entry on oeis.org

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

A338599 Denominators of resistance values < 1 ohm that can be obtained from a network of exactly 9 one-ohm resistors, but not from any network with fewer than 9 one-ohm resistors. Numerators are in A338580.

Original entry on oeis.org

9, 15, 20, 19, 23, 28, 27, 21, 26, 31, 34, 29, 24, 33, 37, 32, 31, 35, 30, 46, 37, 41, 43, 39, 50, 34, 41, 37, 40, 47, 43, 32, 39, 46, 45, 38, 31, 41, 44, 37, 47, 33, 36, 49, 29, 45, 51, 54, 22, 25, 59, 34, 37, 40, 43, 46, 55, 56, 47, 44, 41, 38, 35, 61, 26, 23

Offset: 1

Hugo Pfoertner, Nov 06 2020

			The list of the 894 = A338197(9) resistance values, sorted by increasing size of R = A338580(n)/a(n) = A338609(n)/A338600(9), is the union of [1/9, 2/15, 3/20, ..., 48/49, 50/51, 55/56] and of the corresponding reciprocal resistances > 1 ohm [56/55, 51/50, 49/48, ..., 20/3, 15/2, 9].

Hugo Pfoertner, Table of n, a(n) for n = 1..447

Cf. A338197, A338580, A338600, A338609.

Cf. A338595, A338596, A338597, A338598, A338590 (similar for n = 5..10).

A338596 Denominators of resistance values < 1 ohm that can be obtained from a network of exactly 6 one-ohm resistors, but not from any network with fewer than 6 one-ohm resistors. Numerators are in A338580.

Original entry on oeis.org

6, 9, 11, 10, 11, 13, 12, 9, 11, 13, 13, 11, 9, 12, 13, 11, 10, 11, 9, 13, 10, 11

Offset: 1

Hugo Pfoertner, Nov 05 2020

			The list of the 44 = A338197(6) resistance values, sorted by increasing size of R = A338580(n)/a(n) = A338606(n)/A338600(6), is the union of [1/6, 2/9, 3/11, ..., 11/13, 9/10, 10/11] and of the corresponding reciprocal resistances > 1 ohm [11/10, 10/9, 13/11, ..., 11/3, 9/2, 6].

Cf. A338197, A338580, A338600, A338606.

Cf. A338595, A338597, A338598, A338599, A338590 (similar for n = 5..10)

A338597 Denominators of resistance values < 1 ohm that can be obtained from a network of exactly 7 one-ohm resistors, but not from any network with fewer than 7 one-ohm resistors. Numerators are in A338580.

Original entry on oeis.org

7, 11, 14, 13, 15, 18, 17, 13, 16, 19, 20, 17, 14, 19, 21, 18, 17, 19, 16, 24, 19, 21, 21, 19, 24, 16, 19, 17, 18, 21, 19, 14, 17, 20, 19, 16, 13, 17, 18, 15, 19, 13, 14, 19, 11, 17, 19, 20, 8, 9, 21, 12, 13, 14, 15, 16, 19

Offset: 1

Hugo Pfoertner, Nov 05 2020

			The list of the 114 = A338197(7) resistance values, sorted by increasing size of R = A338580(n)/a(n) = A338607(n)/A338600(7), is the union of [1/7, 2/11, 3/14, ..., 14/15, 15/16, 18/19] and of the corresponding reciprocal resistances > 1 ohm [19/18, 16/15, 15/14, ..., 14/3, 11/2, 7].

Cf. A338197, A338580, A338600, A338607.

Cf. A338595, A338596, A338598, A338599, A338590 (similar for n = 5..10).

A338598 Denominators of resistance values < 1 ohm that can be obtained from a network of exactly 8 one-ohm resistors, but not from any network with fewer than 8 one-ohm resistors. Numerators are in A338580.

Original entry on oeis.org

8, 13, 17, 16, 19, 23, 22, 17, 21, 25, 27, 23, 19, 26, 29, 25, 24, 27, 23, 35, 28, 31, 32, 29, 37, 25, 30, 27, 29, 34, 31, 23, 28, 33, 32, 27, 22, 29, 31, 26, 33, 23, 25, 34, 20, 31, 35, 37, 15, 17, 40, 23, 25, 27, 29, 31, 37, 37, 31, 29, 27, 25, 23, 40, 17, 15

Offset: 1

Hugo Pfoertner, Nov 06 2020

			The list of the 312 = A338197(8) resistance values, sorted by increasing size of R = A338580(n)/a(n) = A338608(n)/A338600(8), is the union of [1/8, 2/13, 3/17, ..., 27/28, 30/31, 34/35] and of the corresponding reciprocal resistances > 1 ohm [35/34, 31/30, 28/27, ..., 17/3, 13/2, 8].

Hugo Pfoertner, Table of n, a(n) for n = 1..156

Cf. A338197, A338573, A338580, A338600, A338608.

Cf. A338595, A338596, A338597, A338599, A338590 (similar for n = 5..10).

cp's OEIS Frontend

A338600 a(n) is the common denominator of the A338197(n) rational resistance values that can be obtained from a network of exactly n one-ohm resistors, but not by a network of fewer than n one-ohm resistors.

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A180414 Number of different resistances that can be obtained by combining n one-ohm resistors.

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A337517 a(n) is the number of distinct resistances that can be produced from a circuit with exactly n unit resistors.

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A338580 Numerators 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. Denominators are in A338590.

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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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A338599 Denominators of resistance values < 1 ohm that can be obtained from a network of exactly 9 one-ohm resistors, but not from any network with fewer than 9 one-ohm resistors. Numerators are in A338580.

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A338596 Denominators of resistance values < 1 ohm that can be obtained from a network of exactly 6 one-ohm resistors, but not from any network with fewer than 6 one-ohm resistors. Numerators are in A338580.

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A338597 Denominators of resistance values < 1 ohm that can be obtained from a network of exactly 7 one-ohm resistors, but not from any network with fewer than 7 one-ohm resistors. Numerators are in A338580.

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A338598 Denominators of resistance values < 1 ohm that can be obtained from a network of exactly 8 one-ohm resistors, but not from any network with fewer than 8 one-ohm resistors. Numerators are in A338580.

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