A339808 -id:A339808 - cp's OEIS frontend

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.

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

A340708 Maximum denominator of resistances obtained by an electrical network with n unit resistors.

Original entry on oeis.org

1, 2, 3, 5, 8, 13, 24, 40, 69, 130, 231, 408, 689, 1272, 2153, 3960, 6993, 12560

Offset: 1

Rainer Rosenthal, Jan 16 2021

a(n) is taken from the set of resistance values counted by A337517(n). These sets can be computed by the PARI program of Andrew Howroyd in A180414.
Also the maximum numerator of these electrical networks for small n.
Maximum numerator and maximum denominator coincide for planar networks: for every resistance R in a planar network with n resistors there is always another planar network with n resistors and resistance 1/R. For nonplanar networks this is not necessarily so, as can be seen in A338573.
The asymmetry is illustrated by the example a(15) = 2153.
The author conjectures that this asymmetry will increase with n, and eventually the maxima will differ.
Conjecture: a(19) = 22233, a(20) = 39918. It would be very desirable to know at which value of n > 18 the maximum values of numerators and denominators differ for the first time. - Hugo Pfoertner, Apr 19 2021

			Denominators for numerator a(15) = 2153 in electrical networks with 15 resistors:
  1025,1049,1051,1058,1089,1104,1145,1184,1185,1193,1208,
  1212,1219,1248,1254,1337,1382,1403,1526,1527,1529,1530,
  1545,1547,1555,1579,1586,1632,1642,1647,1687,1699,1719.
Numerators for denominator a(15) = 2153 in electrical networks with 15 resistors:
   899, 905, 934, 941, 945, 960, 968, 969,1008,1049,1064,
  1095,1102,1104,1128,1137,1143,1147,1164,1182,1207,1296,
  1359,1367,1387,1400,1447,1543.

Index to sequences related to resistances.

Cf. A180414, A337517, A338601, A339548, A339808, A340726.

a(18) from Hugo Pfoertner, Apr 11 2021

cp's OEIS Frontend

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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