A180414 -id:A180414 - cp's OEIS frontend

A338609 Resistance values R < 1 ohm, multiplied by a common denominator 591133442051411133755680800 (= A338600(9)), 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.

65681493561267903750631200, 78817792273521484500757440, 88670016307711670063352120, 93336859271275442171949600, 102805816008941066740118400, 105559543223466273884943000, 109469155935446506251052000, 112596846105030692143939200, 113679508086809833414554000

Offset: 1

Hugo Pfoertner, Nov 06 2020

			The list of the A338197(9)/2 = 447 resistance values < 1 ohm is A338580(n)/A338599(n). a(n) = 591133442051411133755680800 * [1/9, 2/15, 3/20, 3/19, 4/23, 5/28, ..., 43/44, 45/46, 46/47, 48/49, 50/51, 55/56].

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

Cf. A180414, A338197, A338580, A338599, A338600.

Cf. A338605, A338606, A338607, A338608 (similar for n = 5..8).

A342558 a(n) is the maximum number of distinct currents > 0 in a network of n one-ohm resistors with a total resistance of 1 ohm.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 7, 9, 10, 12, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68

Offset: 1

Hugo Pfoertner and Rainer Rosenthal, May 26 2021

nonn

The resistor networks considered here correspond to multigraphs in which each edge is replaced by one or more one-ohm resistors, and in which there are two distinguished nodes, called poles, between which there is a total resistance of 1 ohm.
It was known that the smallest resistor network with all currents being distinct consists of 21 resistors, found by Duijvestin in 1978. This assumes that the network is planar and thus the analogy to the perfectly tiled squares exists, see A014530. For history and references see link to Stuart Anderson's website "SPSS, Order 21".
In 1983, A. Augusteijn and A. J. W. Duijvestijn  described networks in which the number of resistors in a network with distinct resistances was reduced to 20 by allowing the tiled square to be wrapped onto a cylinder. (see links to their publication and to Stuart Anderson's website "Simple Perfect Square-Cylinders")
For values of n greater than 21 increasingly numerous square divisions with a(n) = n exist so that a(n) = n holds for all n > 21 (see A006983).
In the present sequence, networks based on non-planar graphs are allowed, which makes it possible to find networks with a(n) = n also for n = 18 and n = 19.
In the range from n = 13 to n = 17, larger numbers of distinct currents are found than are possible with the methods for generating Mrs. Perkins's quilts, which naturally correspond to planar graphs.

			Examples for n <= 21 are given in the Pfoertner links. Visualizations of tilings corresponding to optimal networks for n <= 12 are given in the Mathworld "Mrs. Perkins's Quilt" link.

Stuart Anderson, Simple Perfect Squared Squares (SPSSs), Order 21 to 37 and higher orders.
Stuart Anderson, SPSS, Order 21.
Stuart Anderson, Simple Perfect Square-Cylinders (SPSCs); Order 20.
Stuart Anderson, Mrs Perkins's Quilt.
A. Augusteijn and A. J. W. Duijvestijn, Simple perfect square-cylinders of low order, Journal of Combinatorial Theory, Series B, Volume 35, Issue 3, December 1983, Pages 333-337
A. J. W. Duijvestijn, Simple perfect squared square of lowest order, Journal of Combinatorial Theory, Series B, Volume 25, Issue 2, October 1978, Pages 240-243
Ed Pegg Jr., List of solutions for the Mrs. Perkins's Quilt Square packing problem.
Hugo Pfoertner, Examples of networks of n one-ohm-resistors with total resistance of 1 ohm, maximizing the number of distinct currents through the single resistors, May 2021, Jan-Apr 2023
Hugo Pfoertner, Visualization of the resistor networks with n >= 13 using the Mathematica graph function, Jan-Apr 2023
Rainer Rosenthal, Cascade graphs for examples n <= 21, January 2023
Rainer Rosenthal, Network and semi-quilt visualizing a(18), January 2023
Eric Weisstein's World of Mathematics, Mrs. Perkins's Quilt.

Cf. A002962, A005670, A006983, A014530, A160911, A180414, A181340, A217156, A337517, A338593, A342556, A360030.

a(n) = n for n >= 18.

A338606 Resistance values R < 1 ohm, multiplied by a common denominator 360360 (= A338600(6)), 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.

Original entry on oeis.org

60060, 80080, 98280, 108108, 131040, 138600, 150150, 160160, 163800, 166320, 194040, 196560, 200200, 210210, 221760, 229320, 252252, 262080, 280280, 304920, 324324, 327600

Offset: 1

Hugo Pfoertner, Nov 05 2020

			The list of resistance values < 1 ohm is A338580(n)/A338596(n). a(n) = 360360 * [1/6, 2/9, 3/11, 3/10, 4/11, 5/13, 5/12, 4/9, 5/11, 6/13, 7/13, 6/11, 5/9, 7/12, 8/13, 7/11, 7/10, 8/11, 7/9, 11/13, 9/10, 10/11].

Cf. A180414, A338197, A338580, A338596, A338600, A338605, A338607, A338608, A338609.

A338999 Number of connected multigraphs with n edges and rooted at two indistinguishable vertices whose removal leaves a connected graph.

Original entry on oeis.org

1, 1, 3, 11, 43, 180, 804, 3763, 18331, 92330, 478795, 2547885, 13880832, 77284220, 439146427, 2543931619, 15010717722, 90154755356, 550817917537, 3421683388385, 21601986281226, 138548772267326, 902439162209914, 5967669851051612, 40053432076016812

Offset: 1

Rainer Rosenthal, Nov 18 2020

nonn

This sequence counts the CDE-descendants of a single edge A-Z.
[C]onnect: different nodes {P,Q} != {A,Z} may form a new edge P-Q.
[D]issect: any edge P-Q may be dissected into P-M-Q with a new node M.
[E]xtend:  any node P not in {A,Z} may form a new edge P-Q with a new node Q.
These basic operations were motivated by A338487, which seemed to count the CDE-descendants of K_4 with edge A-Z removed.

			The a(3) = 3 CDE-descendants of A-Z with 3 edges are
.
         A          A          A
        ( )        /          /
         o        o - o      o - o
         |           /        \
         Z          Z          Z
.
        DCC        DD         DE
.

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, A337517, A338487, A339038, A339045, A339065.

\\ See A339065 for G.
InvEulerT(v)={my(p=log(1+x*Ser(v))); dirdiv(vector(#v,n,polcoef(p,n)), vector(#v,n,1/n))}
seq(n)={my(A=O(x*x^n), g=G(2*n, x+A,[]), gr=G(2*n, x+A,[1])/g, u=InvEulerT(Vec(-1+G(2*n, x+A,[1,1])/(g*gr^2))), t=InvEulerT(Vec(-1+G(2*n, x+A,[2])/(g*subst(gr,x,x^2)))), v=vector(n)); for(n=1, #v, v[n]=(u[n]+t[n]-if(n%2==0,u[n/2]-v[n/2]))/2); v} \\ Andrew Howroyd, Nov 20 2020

a(7)-a(25) from Andrew Howroyd, Nov 20 2020

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

A339548 1 - 1/a(n) is the largest resistance value of this form that can be obtained from a resistor network of not more than n one-ohm resistors.

Original entry on oeis.org

2, 3, 4, 7, 11, 19, 35, 56, 105, 177, 321, 610, 1001, 1893, 3186, 5714, 10073, 18506

Offset: 2

Hugo Pfoertner, Dec 12 2020

			The resistor networks from which the target resistance R = 1 - 1/a(n) can be obtained correspond to simple or multigraphs whose edges are one-ohm resistors. Parallel resistors on one edge are indicated by an exponent > 1 after the affected vertex pair. The resistance R occurs between vertex number 1 and the vertex with maximum number in the graph. In some cases there are other possible representations in addition to the representation given.
.
resistors      vertices
   |     R        |  edges
   2     1/2      2 [1,2]^2
   3     2/3      3 [1,2],[1,3],[2,3]
   4     3/4      4 [1,2],[1,4],[2,3],[3,4]
   5     6/7      4 [1,2]^2,[1,3],[2,4],[3,4]
   6    10/11     5 [1,2],[1,3],[1,4],[2,3],[3,5],[4,5]
   7    18/19     5 [1,2],[1,3]^2,[2,4],[3,4],[3,5],[4,5]
   8    34/35     6 [1,2],[1,3],[1,4],[2,5],[3,4],[3,5],[4,6],[5,6]
   9    55/56     6 [1,2]^2,[1,3],[2,4],[3,5],[3,6],[4,5],[4,6],[5,6]
  10   104/105    7 [1,4],[1,5],[2,4],[2,6],[2,7],[3,5],[3,6],[3,7],[4,6],[5,7]
  11   176/177    7 [1,4],[1,6],[2,4],[2,5],[2,7],[3,5],[3,6],[3,7],[4,6],[4,7],
                    [5,7]
  12   320/321    7 [1,4],[1,6],[2,4],[2,5],[2,6],[2,7],[3,4],[3,5],[3,6],[4,6],
                    [4,7],[5,7]
  13   609/610    8 [1,4],[1,5],[1,7],[2,5],[2,6],[2,7],[3,4],[3,6],[3,7],[4,5],
                    [4,6],[6,8],[7,8]
  14  1000/1001   8 [1,4],[1,5],[1,7],[2,4],[2,5],[2,6],[2,7],[3,5],[3,6],[3,7],
                    [4,5],[4,6],[4,8],[6,8]
  15  1892/1893   9 [1,4],[1,5],[2,5],[2,6],[2,7],[2,9],[3,6],[3,7],[3,8],[3,9],
                    [4,7],[4,8],[4,9],[5,8],[6,8]
  16  3185/3186   9 [1,2],[1,3],[2,6],[2,7],[2,9],[3,6],[3,7],[3,8],[4,5],[4,7],
                    [4,8],[5,6],[5,8],[5,9],[6,7],[8,9]
  17  5713/5714  10 [1,2],[1,3],[2,4],[2,5],[2,7],[3,4],[3,6],[3,10],[4,8],[5,6],
                    [5,7],[5,9],[6,8],[7,8],[7,9],[8,10],[9,10]
  18 10072/10073 10 [1,2],[1,3],[2,4],[2,5],[2,6],[3,4],[3,5],[3,10],[4,8],[5,7],
                    [5,9],[6,7],[6,8],[6,9],[7,8],[7,9],[8,10],[9,10]
  19 18505/18506 11 [1,2],[1,3],[2,5],[2,6],[2,7],[3,4],[3,5],[3,11],[4,6],[4,7],
                    [5,8],[5,10],[6,8],[6,9],[7,9],[7,10],[8,9],[9,11],[10,11]

Fedor Karpelevitch, Program output: lower.txt
Fedor Karpelevitch, Program, run with `./gradlew cir2 --args=best`

Cf. A180414, A337517, A339808.

Cf. A279317, showing that maximum solutions using the square packing analogy can only be obtained for n <= 11 resistors.

a(18) from Hugo Pfoertner, Apr 09 2021

a(19) from Fedor Karpelevitch, Aug 17 2025

A339808 1 + 1/a(n) is the smallest resistance value of this form that can be obtained from a resistor network of not more than n one-ohm resistors.

Original entry on oeis.org

1, 2, 3, 6, 10, 18, 34, 55, 104, 176, 320, 592, 1071, 1855, 3311, 5943, 10231, 19087

Offset: 2

Hugo Pfoertner, Dec 18 2020

			a(2) = 1: 2 resistors in series produce a resistance of 2 = 1 + 1/a(1) ohm.
a(3) = 2: 3 resistors can produce {1/3, 2/3, 3/2, 3} ohms. The smallest resistance > 1 is 3/2 = 1 + 1/a(2) ohms.
a(4) = 3: 4 resistors can produce the A337517(4) = 9 distinct resistances {1/4, 2/5, 3/5, 3/4, 1, 4/3, 5/3, 5/2, 4} of which 4/3 = 1 + 1/a(4) is the smallest resistance > 1 ohm.
a(n) first differs from A339548(n) - 1 for n = 13. The resistance values of the A337517(13) = 110953 distinct resistances that can be obtained from a network of exactly 13 one-ohm resistors closest to 1 ohm are { ..., 551/552, 576/577, 596/597, 609/610, 1, 593/592, 580/579, 552/551, ...}. The largest resistance < 1 of a network of 13 one-ohm resistors is 609/610 = 1 - 1/A339548(13) ohms, whereas the smallest resistance > 1 is 593/a(13) = 593/592 ohms.
The resistor networks from which the target resistance R = 1 + 1/a(n) can be obtained correspond to simple or multigraphs whose edges are one-ohm resistors. Parallel resistors on one edge are indicated by an exponent > 1 after the affected vertex pair. The resistance R occurs between vertex number 1 and the vertex with maximum number in the graph. In some cases there are other possible representations in addition to the representation given.
.
resistors     vertices
   |      R       |   edges
   2     2/1      2  [1,2],[2,3]
   3     3/2      3  [1,2]^2,[2,3]
   4     4/3      4  [1,2]^3,[2,3]
   5     7/6      4  [1,2]^2,[2,3],[2,4],[3,4]
   6    11/10     5  [1,2]^2,[2,3]^2,[2,4],[3,4]
   7    19/18     5  [1,2]^2,[1,3],[2,3],[2,4],[3,5],[4,5]
   8    35/34     6  [1,2]^2,[1,3],[2,3],[2,4],[3,4],[3,5],[4,5]
   9    56/55     6  [1,2],[1,3],[1,4],[2,4],[3,4],[3,5],[4,5],[4,6],[5,6]
  10   105/104    7  [1,3],[1,4],[1,5],[2,4],[2,5],[2,6],[3,4],[3,5],[4,5],
                     [4,6]
  11   177/176    7  [1,2],[1,4],[1,6],[2,6],[2,7],[3,5],[3,6],[3,7],[4,5],
                     [4,6],[5,6]
  12   321/320    7  [1,2],[1,4],[1,5],[2,5],[2,6],[3,5],[3,6],[3,7],[4,5],
                     [4,6],[4,7],[5,6]
  13   593/592    8  [1,4],[1,5],[1,7],[2,4],[2,5],[2,6],[2,7],[3,5],[3,6],
                     [3,7],[3,8],[4,6],[5,8]
  14  1072/1071   9  [1,6],[1,8],[2,7],[2,8],[2,9],[3,5],[3,7],[3,9],[4,6],
                     [4,7],[4,8],[5,6],[5,8],[6,9]
  15  1856/1855   9  [1,5],[1,7],[2,5],[2,6],[2,7],[2,8],[3,6],[3,7],[3,9],
                     [4,6],[4,8],[4,9],[5,8],[5,9],[7,8]
  16  3312/3311  10  [1,7],[1,9],[2,6],[2,7],[2,8],[3,7],[3,8],[3,9],[4,5],
                     [4,6],[4,10],[5,8],[5,9],[6,9],[7,10],[8,10]
  17  5944/5943  10  [1,2],[1,3],[2,4],[2,5],[2,7],[3,4],[3,6],[3,10],[4,6],
                     [4,8],[5,6],[5,8],[6,9],[7,8],[7,9],[8,10],[9,10]
  18 10232/10231 11  [1,2],[1,3],[2,4],[2,6],[2,7],[3,5],[3,8],[3,11],[4,5],
                     [4,9],[5,7],[6,8],[6,9],[7,9],[7,10],[8,10],[9,11],[10,11]
  19 19088/19087 11  [1,2],[1,3],[2,4],[2,5],[2,7],[3,5],[3,6],[3,11],[4,6],
                     [4,9],[5,8],[5,10],[6,7],[6,8],[7,9],[7,10],[8,9],[9,11],[10,11]

Fedor Karpelevitch, Program output: upper.txt
Fedor Karpelevitch, Program, run with `./gradlew cir2 --args=best`

Cf. A180414, A337517, A339548.

a(18) from Hugo Pfoertner, Apr 09 2021

a(19) from Fedor Karpelevitch, Aug 20 2025

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

A340921 a(n) is the number of distinct resistances that can be produced using at most n unit resistors in a planar network.

Original entry on oeis.org

1, 2, 4, 8, 16, 36, 80, 194, 506, 1400, 4024, 11870, 35200, 104836, 311686, 929088, 2776618, 8321128, 24967712, 74985708

Offset: 0

Hugo Pfoertner and Rainer Rosenthal, Feb 14 2021

The relation of this sequence to A340920 is the analog of the relation of A180414 to A337517.

Cf. A153588, A174284, A174286, A180414, A337517, A340920.

a(n) = A180414(n) for n <= 9, a(n) < A180414(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

A338609 Resistance values R < 1 ohm, multiplied by a common denominator 591133442051411133755680800 (= A338600(9)), 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.

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A342558 a(n) is the maximum number of distinct currents > 0 in a network of n one-ohm resistors with a total resistance of 1 ohm.

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A338606 Resistance values R < 1 ohm, multiplied by a common denominator 360360 (= A338600(6)), 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.

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A338999 Number of connected multigraphs with n edges and rooted at two indistinguishable vertices whose removal leaves a connected graph.

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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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A339548 1 - 1/a(n) is the largest resistance value of this form that can be obtained from a resistor network of not more than n one-ohm resistors.

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A339808 1 + 1/a(n) is the smallest resistance value of this form that can be obtained from a resistor network of not more than n one-ohm resistors.

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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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PARI

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A340921 a(n) is the number of distinct resistances that can be produced using at most n unit resistors in a planar network.

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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.