A348020 -id:A348020 - cp's OEIS frontend

A360030 a(n) is the minimum number of equal resistors needed in an electrical network so that n nodes can be selected in this network such that there are n*(n-1)/2 distinct resistances 0 < R < oo between the selected nodes.

1, 3, 5, 8, 10, 11, 12, 14, 15, 16, 18, 19, 21

Offset: 2

Hugo Pfoertner and Rainer Rosenthal, Feb 12 2023

			a(2) = 1, [[1,2]]
.
  1           2
  O----R1R----O
  R_12 = 1
.
a(3) = 3, [[1,2]^2,[2,3]]
.
  1   .---R1R---.   2           3
  O --|         |-- O ---R3R--- O
      .---R2R---.
.
  R_12 = 1/2, R_13 = 3/2,
              R_23 = 1
.
a(4) = 5, node 5 hidden, [[1,2],[2,3]^2,[3,5],[4,5]]
.
  1           2   .---R2R---.   3          (5)          4
  O ---R1R--- O --|         |-- O ---R4R--- O ---R5R--- O
                  .---R3R---.
.
  R_12 = 1, R_13 = 3/2, R_14 = 7/2,
            R_23 = 1/2, R_24 = 5/2,
                        R_34 = 2
.
a(5) = 8, node 6 hidden,
  [[1, 2], [1, 3]^2, [2, 3], [2, 4], [3, 6], [4, 5], [4, 6]]
.
    1             2           4           5
    O-----R1R-----O----R5R----O----R8R----O
    |             |           |
    |            R4R         R7R
    .---R2R---.   |           |
    |         |---O----R6R----O
    .---R3R---.   3          (6)
.
   R_12 = 5/9, R_13 = 7/18, R_14 = 19/18, R_15 = 37/18,
               R_23 = 1/2,  R_24 = 13/18, R_25 = 31/18,
                            R_34 =  8/9,  R_35 = 17/9,
                                          R_45 =  1

IBM Research, Electric networks in graphs, Ponder This Challenge, March 2025, asked for the only network corresponding to a(10)=15 and 4 networks for a(12)=18.
Hugo Pfoertner, Illustrated examples for the terms a(6), a(7), a(8), 17 Feb 2023.
Hugo Pfoertner, Illustrated examples for the terms a(9), a(10), a(11), 3 Apr 2023.
Hugo Pfoertner, Illustration of a(12)=18, 8 Jan 2024, showing 3 planar and 5 non-planar networks, 4 of which were required to solve the bonus question of IBM's Ponder This Challenge.
Hugo Pfoertner and Klaus Nagel, Illustration of a(14)=21, 21 Aug 2025.

Cf. A219158, A342557, A342558, A348020.

a(14) from Klaus Nagel and Hugo Pfoertner, Aug 21 2025

A347282 a(n) is the least number of unit resistors in an electrical network with total resistance A007305(n)/A007306(n) such that all currents through the resistors are distinct and > 0.

Original entry on oeis.org

18, 19, 20, 18, 26, 19, 17, 19

Offset: 1

Hugo Pfoertner, Oct 12 2021

The trivial case of a single resistor is excluded, thus making a(1) = A342558(18) = 18.

			  n  A007305(n)/ a(n)   Edge list of network graph (example)
     A007306(n)
  1     1/1       18    [[1, 5], [1, 6], [1, 7], [2, 5], [2, 7], [2, 8],
                         [3, 6], [3, 7], [3, 8], [3, 9], [10, 6], [4, 8],
                         [4, 9], [5, 8], [5, 9], [6, 9], [7, 9], [10, 4]]
                        Poles: [5, 10]
  2     1/2       19    see linked file for description of solutions
  3     1/3       20
  4     2/3       18
  5     1/4       26
  6     2/5       19
  7     3/5       17
  8     3/4       19
  9     1/5    unknown

Hugo Pfoertner, Examples of networks corresponding to a(1)-a(8), October 2021.
Index to sequences related to resistances.

Cf. A007305, A007306, A342558, A348020.

cp's OEIS Frontend

A360030 a(n) is the minimum number of equal resistors needed in an electrical network so that n nodes can be selected in this network such that there are n*(n-1)/2 distinct resistances 0 < R < oo between the selected nodes.

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